BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Evgeny Khukhro (University of Lincoln) DTSTART:20210325T161000Z DTEND:20210325T164000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/1 DESCRIPTION:Title: Compact groups with countable Engel sinks\nby Evgeny Khukhro ( University of Lincoln) as part of Ischia Group Theory\, a.k.a. The 24 Hour s of GOThIC (a 24-hour conference)\n\n\nAbstract\nAn Engel sink of an elem ent $g$ of a group $G$ is a set ${\\mathscr E}(g)$ such that for every $x\ \in G$ all sufficiently long commutators $[...[[x\,g]\,g]\,\\dots \,g]$ be long to ${\\mathscr E}(g)$. (Thus\, $g$ is an Engel element precisely whe n we can choose ${\\mathscr E}(g)=\\{ 1\\}$.) It is proved that if every e lement of a compact (Hausdorff) group $G$ has a countable (or finite) Enge l sink\, then $G$ has a finite normal subgroup $N$ such that $G/N$ is loca lly nilpotent. This settles a question suggested by J.~S.~Wilson.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Agata S. Atkarskaya (Bar-Ilan University) DTSTART:20210325T165000Z DTEND:20210325T173000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/2 DESCRIPTION:Title: Combinatorial approach for Burnside groups of relatively small odd exponents\nby Agata S. Atkarskaya (Bar-Ilan University) as part of Is chia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n \n\nAbstract\nLet $B(m\, n)$ be the free Burnside group with $m \\geqslant 2$ free generators and the identity $x^n = 1$. Burnside problem asks for which positive integers $m\, n$ the free Burnside group $B(m\, n)$ is infi nite. We study this question for odd exponents $n$. There are two well-kno wn approaches to study these groups. The first is a combinatorial approach that was introduced by P.\\\,Novikov and S.\\\,Adian (1968). They proved that $B(m\, n)$ is infinite for odd $n \\geqslant 665$ (1975). The second one is a geometric approach that was introduced by Yu.\\\,Olshansky (1982) . Yu.\\\,Olshansky proved that $B(m\, n)$ is infinite for odd $n > 10^{10} $. However\, his proof is much more transparent then the one of P.\\\,Novi kov and S.\\\,Adian. There is also a new approach by T.\\\,Delzant and M.\ \\,Gromov (2008)\, see also R.\\\,Coulon (2013). Their approach is based o n advanced theory of metric spaces and works only for a very large exponen t~$n$.\n\nIn general our approach is combinatorial\, based on iterated sma ll cancellation theory. It allows us to decrease significantly the exponen t $n$ for which $B(m\, n)$ is infinite. The main instrument that we are us ing is a special choice of canonical (in a sense of Rips) representatives of cosets of $B(m\, n)$. The principal property of the canonical represent atives is as follows. It is well known that if we consider a product of tw o words in a free group\, their cancellation tree is a tripod. Let $a\, b \\in B(m\, n)$ and $ \\mathrm{can}(a)\, \\mathrm{can}(b)$ be the canonical representatives. Then $ \\mathrm{can}( \\mathrm{can} \\cdot \\mathrm{can }(b))$ can be graphically represented as a graph that is very close to a t ripod in a certain sense. This allows to establish small cancellation prop erties at each iteration step.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Waldemar Holubowski (Silesian University of Technology) DTSTART:20210325T180500Z DTEND:20210325T183500Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/3 DESCRIPTION:Title: Normal subgroups in the group of column-finite infinite matrices\nby Waldemar Holubowski (Silesian University of Technology) as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference) \n\n\nAbstract\n(Joint work with M. Maciaszczyk and S. Zurek.)\n\nThe clas sical result\, due to Jordan\, Burnside\, Dickson\, says that every normal subgroup of $GL(n\, K)$ ($K$ - a field\, $n \\geq 3$) which is not contai ned in the center\, contains $SL(n\, K)$. A. Rosenberg gave description of normal subgroups of $GL(V)$\, where $V$ is a vector space of any infinite cardinality dimension. However\, when he considers subgroups of the group of linear transformations $g$ such that $g-id_V$ has finite dimensional ranges the proof is not complete. We fill this gap for countably dimension al $V$ giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any f ield. Similar results for Lie algebras are also proven.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Aleksandr Olshansky (Vanderbilt University) DTSTART:20210325T184000Z DTEND:20210325T191000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/4 DESCRIPTION:Title: Groups Finitely Presented in Burnside Varieties\nby Aleksandr Olshansky (Vanderbilt University) as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\n\nAbstract\nFor all suffi ciently large odd integers $n$\, the following version of Higman's embeddi ng theorem is proved in the variety ${\\cal B}_n$ of all groups satisfying the identity $x^n=1$. A finitely generated group $G$ from ${\\cal B}_n$ h as a presentation $G=\\langle A\\mid R\\rangle$ with a finite set of gener ators $A$ and a recursively enumerable set $R$ of defining relations if an d only if it is a subgroup of a group $H$ finitely\npresented in the varie ty ${\\cal B}_n$. This answers S.V. Ivanov's question raised in 1992. It f ollows that there is a $2$-generated finitely presented in ${\\cal B}_n$ group containing isomorphic copies of all finitely presented in ${\\cal B} _n$ groups as subgroups.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Delaram Kahrobaei (University of York and New York University) DTSTART:20210326T152000Z DTEND:20210326T155000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/5 DESCRIPTION:Title: Algorithmic problems in Engel groups and cryptographic application s\nby Delaram Kahrobaei (University of York and New York University) a s part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour c onference)\n\n\nAbstract\n(Joint work with Marialaura Noce\, Antonio Torto ra and Maria Tota.)\n\nThe theory of Engel groups plays an important role in group theory since these groups are closely related to the Burnside pro blems. In this survey talk we consider several classical and novel algorit hmic problems for Engel groups and propose several open problems. We study these problems with a view towards applications to cryptography. This pat h of research has been initiated by a visit to University of Salerno invit ed by Antonio Tortora and Maria Tota. I am presenting joint work with them and Marialaura Noce.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Pavel Shumyatsky (University of Brasilia) DTSTART:20210325T202000Z DTEND:20210325T205000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/6 DESCRIPTION:Title: On profinite groups with restricted centralizers of commutators\nby Pavel Shumyatsky (University of Brasilia) as part of Ischia Group Th eory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\n\nAbstract\ n(Joint work with Eloisa Detomi and Marta Morigi.)\n\nA group $G$ has res tricted centralizers if for each $g$ in $G$ the centralizer $C_G(g)$ eithe r is finite or has finite index in $G$. A theorem of Shalev states that a profinite group with restricted centralizers is abelian-by-finite. In this talk I will describe the theorem that if $w$ is a multilinear commutator word and $G$ a profinite group with restricted centralizers of $w$-values\ , then the verbal subgroup $w(G)$ is abelian-by-finite. This is a joint wo rk with Eloisa Detomi and Marta Morigi.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Emanuele Pacifici (University of Milan) DTSTART:20210325T220000Z DTEND:20210325T223000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/7 DESCRIPTION:Title: On Huppert's Rho-Sigma conjecture\nby Emanuele Pacifici (Unive rsity of Milan) as part of Ischia Group Theory\, a.k.a. The 24 Hours of GO ThIC (a 24-hour conference)\n\n\nAbstract\n(Joint work with Z. Akhlaghi an d S. Dolfi)\n\nThe set \\({\\rm{cd}}(G)=\\{\\chi(1)\\mid\\chi\\in{\\rm{Irr }}(G)\\}\\) consisting of the degrees of the irreducible complex character s of a finite group \\(G\\) has been an object of considerable interest si nce the second part of the \\(20^{\\rm{th}}\\) century\, and the study of the arithmetical structure of this set is a particularly intriguing aspect of Character Theory of finite groups (see for instance \\cite{L}). \nA r emarkable question in this research area was posed by Bertram Huppert in t he 80's: is it true that at least one of the character degrees is divis ible by a ``large" portion of the entire set of primes that appear as di visors of some character degree?\nMore precisely\, denoting by $\\pi(n)$ the set of prime divisors of an integer $n$\, and writing for short\n$\\pi (\\chi)$ instead of $\\pi(\\chi(1))$ when $\\chi \\in {\\rm{Irr}}(G)$\, on e defines\n%\n$$\\rho(G) = \\bigcup_{\\chi \\in {\\rm{irr}}(G)} \\pi(\\chi )$$\nand\n$$ \\sigma(G) = \\max \\{ |\\pi(\\chi)| \\mid \\chi \\in {\\rm{i rr}}(G) \\}\;$$\n\n\\noindent Huppert's \\emph{$\\rho$-$\\sigma$ conjectur e} predicts that $|\\rho(G)| \\leq 3 \\sigma(G)$ holds for every finite g roup $G$\,\nand that $|\\rho(G)| \\leq 2 \\sigma(G)$ if $G$ is solvable.\n It is worth noting that the bounds are in some sense best possible\, as t hey are attained for the groups $A_5$ and $S_4$\, respectively.\n\nDuring the last four decades\, several contributions have been given toward the p roof of this conjecture.\nFor solvable groups\, the conjecture was proved true by D. Gluck (\\cite{G}) for $\\sigma(G) \\leq 2$\, and also in the ca se that all degrees in ${\\rm{cd}}(G)$ are square-free numbers. The best b ound known till now was obtained by O. Manz and T.R. Wolf\; they proved in ~\\cite{MW0} that\, if $G$ is solvable\, then $|\\rho(G)| \\leq 3 \\sigma (G) +2$.\n\nAs for the non-solvable case\, the $\\rho$-$\\sigma$ conjectur e was proved true for all finite non-abelian simple groups by D.L. Alvis a nd M. Barry (\\cite{AB})\, whereas the general but weaker bound $|\\rho(G) | \\leq 7 \\sigma(G)$ was obtained by C. Casolo and S. Dolfi in~\\cite{CD} .\n\nIn this talk\, we will present some recent developments in the study of Huppert's conjecture (\\cite{ADP})\, including an improvement of Manz a nd Wolf's theorem for solvable groups.\n\n\n\\begin{thebibliography}{99}\n \n\\bibitem{ADP} Z. Akhlaghi\, S. Dolfi\, E. Pacifici\, \\emph{On Huppert' s Rho-Sigma conjecture}\, submitted.\n\n\\bibitem{AB} D.L. Alvis\, M. Barr y\, \\emph{Character degrees of simple groups}\, J. Algebra 140 (1991)\, 1 16--123.\n\n\\bibitem{CD} C. Casolo\, S. Dolfi\, \\emph{Prime divisors of irreducible character degrees and of conjugacy class sizes in finite group s}\, J. Group Theory 10 (2007)\, 571--583. \n\n\\bibitem{G} D. Gluck\, \\e mph{A conjecture about character degrees of solvable groups}\, J. Algebra 140 (1991)\, 26--35. \n\n\\bibitem{L} M.L. Lewis\, \\emph{An overview of g raphs associated with character degrees and conjugacy class sizes in finit e groups}\, Rocky Mountain J. Math. 38 (2008)\, 175--211.\n\n\\bibitem{MW0 } O. Manz\, T. Wolf\, \\emph{Arithmetically long orbits of solvable linear groups}\, Illin. J. Math. 37 (1993)\, 652--665.\n\n\\end{thebibliography} \n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Luise-Charlotte Kappe (University of Binghmapton) DTSTART:20210325T212000Z DTEND:20210325T215000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/8 DESCRIPTION:Title: Finite Coverings of Semigroups\nby Luise-Charlotte Kappe (Univ ersity of Binghmapton) as part of Ischia Group Theory\, a.k.a. The 24 Hour s of GOThIC (a 24-hour conference)\n\n\nAbstract\nFor a group\, $G$\, the covering number of $G$ with respect to subgroups\, $\\sigma_g(G)$\, is the minimum number of proper subgroups of $G$ whose union is $G$. Covering n umbers of groups have been well-studied\, and it is known whether there ex ists a group $G$ such that $\\sigma_g(G)=n$ for each $n$ between 2 and 129 . For example\, no group has covering number 2\, 7\, or 11. Recently\, w e have explored covering numbers of semigroup. We denote the covering num ber of a semigroup $S$ with respect to semigroups by $\\sigma_s(S)$. This analagous problem produces quite different results. Specifically\, for a finite semigroup $S$ that is neither a group nor monogenic\, we have $\\si gma_s(S)=2$. Our main result gives a complete description of $\\sigma_s(S )$ when $S$ is a finite semigroup (modulo groups). We also have partial r esults for some infinite cases such as when $S$ is a monoid or a group. L astly\, we will present theorems describing covering inverse semigroups wi th inverse subsemigroups and covering monoids with submonoids.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Pablo Spiga (University of Milan-Bicocca) DTSTART:20210325T225000Z DTEND:20210325T232000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/9 DESCRIPTION:Title: Milnor-Wolf's Theorem for group endomorphisms\nby Pablo Spiga (University of Milan-Bicocca) as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\n\nAbstract\nFor a group $G$\, denote by $\\mathcal F(G)$ the family of all finite non-empty subsets of $G$. If $\\phi:G\\to G$ is an endomorphism and $F\\in\\mathcal F(G)$\, the \\emph{growth function} of $\\phi$ with respect to $F$ is \n$$\\gamma_{\ \phi\,F}:\\\,\\mathbb{N}\\to\\mathbb{N}\\\\\nn\\mapsto |T_n(\\phi\,F)|\,\n $$\nwhere \n$$T_n(\\phi\,F):=F\\phi(F)\\cdots\\phi^{n-1}(F)\n:=\\{f_0\\phi (f_1)\\cdots\\phi^{n-1}(f_{n-1}):f_0\,\\ldots\,f_{n-1}\\in F\\}.\n$$\nBroa dly speaking\, this definition generalises ``dynamically'' the classic def inition of word growth in finitely generated groups.\n\nIn this talk we di scuss some recent results on the growth of group endomorphisms\; in partic ular\, we discuss an analogue of Chou's extension of Milnor-Wolf's theorem for finitely generated groups. \n$$\\text{If $G$ is an elementary amenabl e group and $\\phi:G\\to G$ an endomorphism\,\n then $\\phi$ has either po lynomial or exponential growth.}$$During the talk\, we include some proble ms that have naturally arisen in this work.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Igor Subbotin (National University\, Los Angeles) DTSTART:20210325T233000Z DTEND:20210326T000000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/10 DESCRIPTION:Title: Methods of Group Theory in Leibniz Algebras: Some Compelling Resu lts\nby Igor Subbotin (National University\, Los Angeles) as part of I schia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\ n\n\nAbstract\nWe will observe some important\, recent results in Leibniz algebras obtained by applying methods and approaches that were developed i n group theory\, especially the ones that have proven to be effective in t he study of groups. We cannot talk about direct similarity of results\, bu t we will highlight the viability of this approach.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Mark Sapir (Vanderbilt University) DTSTART:20210326T002000Z DTEND:20210326T005000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/11 DESCRIPTION:Title: On closed subgroups of the R. Thompson group $F$\nby Mark Sap ir (Vanderbilt University) as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\n\nAbstract\nWith every subgroup $H$ of $F$ one can associate a possibly bigger subgroup $\\bar H\\ge H$ wh ich is called the closure of $H$. This operation satisfies the usual closu re properties and subgroups $H$ with $H=\\bar H$ are called closed. By a r esult of Golan-Polak\, $H$ is closed iff every function from $F$ which is locally from $H$ is in $H$. For example\, the group $F$ itself\, centraliz ers of elements of $F$\, many solvable subgroups of $F$ and stabilizers of points on $[0\,1]$ are closed subgroups of $F$\n\nOne can also characteri se closed subgroups of $F$ as diagram groups of tree semigroup presentatio ns. Where a semigroup presentation $\\mathcal{P}=\\{X|u_i=v_i\\}$ is calle d a tree presentation if $|u_i|=2$\, $|v_i|=1$ for every $i$ and if for so me $i\,j$ $u_i=u_j$ or $v_i=v_j$\, then $i=j$. For example $F$ is the diag ram group of the presentation $\\{x| xx=x\\}$ .\n\nIf $\\mathcal{P}$ is fi nite then we say that $H=\n\\bar H$ is finitely given. For example the clo sure of any finitely generated subgroup of $F$ is finitely given. A finite ly given subgroups of $F$ are similar to finitely generated subgroups of f ree groups. For example the membership problem for such a subgroup is easi ly decidable.\n\nWe construct finitely given closed subgroups of $F$ with undecidable conjugacy problem. That fact follows from the result that ever y finitely presented group can be given by a finite tree presentation.\n\n We also prove that many finitely generated closed subgroups of $F$ are und istorted.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Mark Lewis (Kent State University) DTSTART:20210326T010000Z DTEND:20210326T014000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/12 DESCRIPTION:Title: Graphs associated with groups\nby Mark Lewis (Kent State Univ ersity) as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\n\nAbstract\nThere are many different graphs that ha ve been associated\nwith groups. In our talk\, we consider the commuting graph and several\nrelated graphs. In particular\, we will survey the kno wn results\nregarding these graphs. We will present a new result that ext ends to\nParker's result on the commuting graphs of centerless solvable gr oups.\nWe present an alternate characterization of the commuting graph in terms\nof centralizers. We will present some new results on the cyclic gr aph\n(a.k.a. the enhanced power graph)\, and some new results on the solub le\ngraph.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Efim Zelmanov (University of California\, San Diego) DTSTART:20210326T020000Z DTEND:20210326T023000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/13 DESCRIPTION:Title: Growth functions of groups\, algebras and monoids\nby Efim Ze lmanov (University of California\, San Diego) as part of Ischia Group Theo ry\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\n\nAbstract\nW e will discuss how infinite algebraic structures grow (growth functions) a nd\nhow complicated they are (Dehn functions).\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Organisers DTSTART:20210325T223500Z DTEND:20210325T224500Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/14 DESCRIPTION:Title: A virtual tour of La Mortella gardens\nby Organisers as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conferen ce)\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Organisers DTSTART:20210325T173500Z DTEND:20210325T175000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/15 DESCRIPTION:Title: A virtual tour of Ischia\nby Organisers as part of Ischia Gro up Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\nAbstra ct: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Organisers DTSTART:20210325T175000Z DTEND:20210325T180500Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/16 DESCRIPTION:Title: Remembering Kanta Gupta\nby Organisers as part of Ischia Grou p Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\nAbstrac t: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Marino Cogliani DTSTART:20210325T191500Z DTEND:20210325T194000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/17 DESCRIPTION:Title: Intermezzo: Neapolitan Songs\nby Marino Cogliani as part of I schia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\ n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Sandra Castellano and Rebeca Ferri DTSTART:20210325T205500Z DTEND:20210325T212000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/18 DESCRIPTION:Title: Intermezzo: Flute and Cello Concerto\nby Sandra Castellano an d Rebeca Ferri as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOT hIC (a 24-hour conference)\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Giudici (University of Western Australia) DTSTART:20210326T034000Z DTEND:20210326T041000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/19 DESCRIPTION:Title: Automorphism orbits of groups and the Monster\nby Michael Giu dici (University of Western Australia) as part of Ischia Group Theory\, a. k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\n\nAbstract\nThe orde r of an element of a group is a natural invariant of an automorphism. In 1992\, Zhang characterised all finite groups such that for all integers $k $ the automorphism group acts transitively on the set of all elements of order $k$. Such groups are called AT-groups. In this talk\, I will discuss recent joint work with Alexander Bors and Cheryl Praeger that investigate s two measures of how close a group is to being an AT-group. This includes a new interesting characterisation of the Monster simple group.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Organisers DTSTART:20210326T031500Z DTEND:20210326T033000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/20 DESCRIPTION:Title: A virtual tour of Ischia\nby Organisers as part of Ischia Gro up Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\nAbstra ct: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Stephen Glasby (University of Western Australia) DTSTART:20210326T024000Z DTEND:20210326T031000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/21 DESCRIPTION:Title: Most permutations power to a cycle of small prime length\nby Stephen Glasby (University of Western Australia) as part of Ischia Group T heory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\n\nAbstract \nA permutation $g\\in\\textup{Sym}(n)$ is called a \\emph{pre-$p$-cycle} if\nsome power of $g$\, say $g^r$\, is a $p$-cycle (that is\, $g^r$ has on e $p$-cycle\nand $n-p$ fixed points).\nMotivated by probabilistic algorith ms in computational group theory\nwe prove:\nthat most permutations of $\\ textup{Alt}(n)$ or $\\textup{Sym}(n)$ are\npre-$p$-cycles for some prime $ p$ in a very small range\n$\\log nA strong version of the Sims conjecture on finite primitive perm utation groups\nby Anatoly Kondrat'ev (Ural Branch of the Russian Acad emy of Sciences) as part of Ischia Group Theory\, a.k.a. The 24 Hours of G OThIC (a 24-hour conference)\n\n\nAbstract\nThe well-known Sims conjecture on finite primitive permutation groups\ncan be formulated using graphs a s follows. For an undirected connected graph $\\Gamma$ with\nvertex set $V (\\Gamma)\,$\\ $G \\leq Aut(\\Gamma)$\,\\ $x\\in V(\\Gamma)$\, and\n$i\\in {\\mathbb N}\\cup \\{0\\}$\, denote by $G_x^{[i]}$ the elementwise stabil izer in~$G$ of the ball of \nradius~$i$ of the graph~$\\Gamma$ centered at ~$x$ in the natural metric~on~$V(\\Gamma)$. Then the Sims conjecture is \n equivalent to the following statement: There exists a function\n$\\psi: \\ mathbb N\\cup\\{0\\}\\longrightarrow \\mathbb N$ such that\, if\\\, $\\Gam ma$ is an\nundirected connected finite graph and~$G$ is its automorphism g roup acting primitively on\n$V(\\Gamma)$\, then $G_x^{[\\psi(d)]}=1$ for $ x\\in V(\\Gamma)$\, where $d$ is the valency of the graph~$\\Gamma$.\nThe validity of the Sims conjecture was proved by P.J. Cameron\, C.E. Praeger\ , J. Saxl\, G.M. Seitz in \n[Bull. London Math. Soc. 15 (1983)\, no. 5\, 4 99--506].\n\nIn [Dokl. Math. 59 (1999)\, no. 1\, 113-115]\, the speaker an d V.I. Trofimov obtained the following strengthened version \nof the Sims conjecture: If $\\Gamma$ is an undirected connected finite graph and $G$ i s its automorphism group acting \nprimitively on $V(\\Gamma)$\, then $G_x ^{[6]}=1$ for $x\\in V(\\Gamma)$. Note that there exist $\\Gamma$ and $G$ acting \nprimitively on $V(\\Gamma)$ such that $G_x^{[5]}\\not=1$ for $x\\ in V(\\Gamma)$.\n\nNow we investigate the more general problem of describi ng all pairs $(\\Gamma\, G)$\, where $\\Gamma$ is an undirected \nconnect ed finite graph\, $G$ is an automorphism group of $\\Gamma$ acting primi tively on $V(\\Gamma)$ and \n$G_x^{[2]}\\not=1$ for $x\\in V(\\Gamma)$. \n \nIn the talk\, we discuss our results concerning this problem.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Marino Cogliani DTSTART:20210326T045500Z DTEND:20210326T051000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/23 DESCRIPTION:Title: Intermezzo: Neapolitan Songs\nby Marino Cogliani as part of I schia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\ n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Alireza Abdollahi (University of Isfahan) DTSTART:20210326T051000Z DTEND:20210326T054000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/24 DESCRIPTION:Title: Compact groups with a set of positive Haar measure satisfying a n ilpotent law\nby Alireza Abdollahi (University of Isfahan) as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference) \n\n\nAbstract\nThe following question is proposed as Question 1.20 of [A. Martino\, M. C. H. Tointon\, M. Valiunas and E. Ventura\, \nProbabilist ic Nilpotence in infinite groups\, to appear in Israel J. Math.]:\\\\\nLe t $G$ be a compact group\, and suppose that $$\\mathcal{N}_k(G) = \\{(x_1\ ,\\dots\,x_{k+1}) \\in G^{k+1} \\\;|\\\;\n[x_1\,\\dots\, x_{k+1}] = 1\\}$$ has positive Haar measure in $G^{k+1}$. Does $G$ have\nan open $k$-step n ilpotent subgroup?\\\\\nThe case $k = 1$ is already known [K. H. Hofmann a nd F. G. Russo\, The probability that $x$ and $y$ commute in a compact gro up\,\nMath. Proc. Cambridge Philos. Soc. 153 (2012)\, no. 3\, 557-571.]. W e positively answer the above question for $k = 2$.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Nikolai Vavilov (Saint Petersburg State University) DTSTART:20210326T055000Z DTEND:20210326T062000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/25 DESCRIPTION:Title: The Yoga of commutator revisited\, revisited\nby Nikolai Vavi lov (Saint Petersburg State University) as part of Ischia Group Theory\, a .k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\n\nAbstract\n(Joint talk with\n Zuhong ZHANG\,\n Beijing Institute of Technology.)\n\nTh e talk is an overview of a series of our joint recent papers on\ncommutato r formulas in classical groups and Chevalley groups.\n\n Let $A$ and $ B$ be two ideals of a commutative ring $R$. The study\nof commutator subgr oups\n$$[\\mathrm{GL}(n\,R\,A)\,GL(n\,R\,B)]\, \\qquad [\\mathrm{GL}(n\,R\ ,A)\,E(n\,R\,B)]\, \\qquad [\\mathrm{E}(n\,R\,A)\, \\mathrm{E}(n\,R\,B)]$$ \nand other related birelative groups has a venerable history. It goes\nba ck to the beginnings of algebraic $K$-theory in the works of Bass\nand oth er classics.\n\n After a brief historical survey we plan to outline the very recent\nunexpected developments\, including unrelativised commutator \nformulas\, generation results\, and some applications.\n\n Recently we proved yet another amazing result in this direction\,\nasserting that ins tead of getting rid of all elementary commutators\nexcept one\, we can get rid of all elementary conjugates\, leaving\nonly elementary commutators a s generators.\n\n We plan also to mention also generalisations to unitar y groups\,\nand to Chevalley groups and some further prospects.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Sandra Castellano and Rebeca Ferri DTSTART:20210326T062500Z DTEND:20210326T065000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/26 DESCRIPTION:Title: Intermezzo: Flute and Cello Concerto\nby Sandra Castellano an d Rebeca Ferri as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOT hIC (a 24-hour conference)\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Sandra Castellano and Rebeca Ferri DTSTART:20210326T015000Z DTEND:20210326T020000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/27 DESCRIPTION:Title: Intermezzo: Flute and Cello Concerto\nby Sandra Castellano an d Rebeca Ferri as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOT hIC (a 24-hour conference)\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Mahmut Kuzucuoğlu (Middle East Technical University) DTSTART:20210326T065000Z DTEND:20210326T072000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/28 DESCRIPTION:Title: Explicit Examples of Algebraically Closed Groups\nby Mahmut K uzucuoğlu (Middle East Technical University) as part of Ischia Group Theo ry\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\n\nAbstract\nL et $W_i(x_j\, g_k)$ be a word in indeterminates $x_j$ where $ \\ j=1\, \\l dots \,n$ and $g_k\\in G$ for $k=1\, \\ldots \,m$. A group $G$ is called a lgebraically closed if and only if for every finite set\n$$W_i(x_j\, g_k)= 1 \\hspace{2.cm} (i=1\,2 \\ldots l)$$\n$$ W_i(x_j\, g_k)\\neq 1 \\hspace{ 2.cm} (i=l+1\,\\ldots \,s)$$\nof equations and inequations which is consis tent with $G$ already has a solution in $G$. Recall that the above system of equations and inequations is called consistent with $G$ if there exists a group $H$ and a monomorphism $\\psi:G\\rightarrow H$ such that the abov e system has a solution in $H$.\nAlgebraically closed groups first introdu ced and studied by W. R. Scott in \\cite{Scott51}. B. H. Neumann in \\ci te{Neumann73} stated that ``However\, no algebraically closed group is ex plicitly known\, the existence proof being highly non-constructive. This s tem in part from the fact that there is no useful criterion known that tel ls one what sentences are or are not consistent over a given group''. In t his talk we will give as is stated in \\cite{kakeku} explicit examples of algebraically closed groups for large cardinals. Hence answer Neumann's qu estion positively.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Carmen Musella (University of Naples ``Federico II'') DTSTART:20210326T073000Z DTEND:20210326T080000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/29 DESCRIPTION:Title: Generalized nilpotency properties for subgroup lattices of groups \nby Carmen Musella (University of Naples ``Federico II'') as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference) \n\n\nAbstract\nA rather natural way for trying to obtain a lattice-theore tic characterization of a class of groups $\\mathfrak{X}$ is to replace th e concepts appearing in the\ndefinition of $\\mathfrak{X}$ by lattice-theo retic concepts. The first to use this idea were\nKantorovic and Plotkin wh o in 1954 introduced the notion of a modular\nchain in a lattice\, as tran slation of a central series of a group\, to determine a lattice-theoretic characterization of the class of torsion-free nilpotent\ngroups.\n\nIn thi s talk\, we will present new applications of this translation method\nto s tudy from a lattice point of view some classes of infinite groups with\nge neralized nilpotency properties.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Organisers DTSTART:20210326T080500Z DTEND:20210326T082000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/30 DESCRIPTION:Title: Remembering Zvi Arad\nby Organisers as part of Ischia Group T heory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Marcel Herzog (Tel Aviv University) DTSTART:20210326T082000Z DTEND:20210326T085000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/31 DESCRIPTION:Title: New criteria for solvability\, nilpotency and other properties of finite groups\nby Marcel Herzog (Tel Aviv University) as part of Isch ia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\n \nAbstract\nLet $G$ denote a finite group. We shall describe a series of n ew results\nwhich determine criteria for $G$ being solvable\, nilpotent o r having \nother properties\, based either on the orders of the elements of $G$ or \non the orders of the subgroups of $G$. In particular\, we shal l describe\nthe following new result of Patrizia Longobardi\, Mercede Maj and myself:\nIf the summation of $|H|/|G|$ over all subgroups $H$ of $G$ i s less \nthen $117/20$\, then $G$ is solvable. This result is best possibl e\, since \nfor $G=A_5$ the above summation equals $117/20$.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikhail E. Muzychuk (Ben Gurion University of the Negev) DTSTART:20210326T090000Z DTEND:20210326T093000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/32 DESCRIPTION:Title: On Some Applications of Group Representation Theory to Algebraic Problems Related to the Congruence Principle for Equivariant Maps\nby Mikhail E. Muzychuk (Ben Gurion University of the Negev) as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\n\n Abstract\n(Dedicated to the memory of Zvi Arad.)\n\nGiven a finite group $ G$ and two unitary $G$-representations\n$V$ and $W$\, possible restriction s on Brouwer degrees of\nequivariant maps between representation spheres $ S(V)$ and $S(W)$\nare usually expressed in a form of congruences modulo t he\ngreatest common divisor of lengths of orbits in $S(V)$\n(denoted $\\a lpha(V)$).\nEffective applications of these congruences are limited by\nan swers to the following questions:\n(i) under which conditions\, is $\\alph a(V)>1$?\nand (ii) does there exist an equivariant map with the degree\nea sy to calculate?\n\nIn my talk I'll present a number of results about the parameter $\\alpha(V)$. One of them proves that a finite group $G$ is solv able if and only if \n$\\alpha(V)>1$ for each irreducible non-trivial\n$\\ mathbb{C}[G]$-module $V$.\nThis provides a new solvability criterion for f inite groups.\n\nRegarding the second question\, we suggest a class\nof No rton algebras without 2-nilpotents giving\nrise to equivariant quadratic m aps\, which\nadmit an explicit formula for the Brouwer degree.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Sandra Castellano and Rebeca Ferri DTSTART:20210326T094000Z DTEND:20210326T095000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/33 DESCRIPTION:Title: Intermezzo: Flute and Cello Concerto\nby Sandra Castellano an d Rebeca Ferri as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOT hIC (a 24-hour conference)\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Leonid Andreevich Kurdachenko (Oles Honchar Dnipro National Univer sity) DTSTART:20210326T095000Z DTEND:20210326T102000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/34 DESCRIPTION:Title: Around Contranormality\nby Leonid Andreevich Kurdachenko (Ole s Honchar Dnipro National University) as part of Ischia Group Theory\, a.k .a. The 24 Hours of GOThIC (a 24-hour conference)\n\n\nAbstract\nA periodi c group $G$ is said to be Sylow--nilpotent if $G$ is locally nilpotent a nd the Sylow $p$-subgroup of $G$ is nilpotent for each prime $p$. Ever y Sylow--nilpotent group does not include proper contranormal subgroups. W e show some results about the groups without proper contranormal subgroups . \n\n Theorem 1. Let $G$ be a periodic group\, $H$ be a normal locally nilpotent subgroups of $G$ such that $G/H$ is nilpotent. Moreover\, if $H$ is nilpotent and $\\Pi(H) \\cap \\Pi(G/H) = \\emptyset$\, then $G $ is nilpotent.\n\n Theorem 2. et $G$ be a locally finite group and $H$ be a normal locally nilpotent subgroup such that the Sylow $p$-subgroups of $G/H$ are Chernikov for all prime $p$. If $G$ does not include pro per contranormal subgroup\, then $G$ is locally nilpotent.\n\n Theorem 3 . Let $G$ be a locally generalized radical group\, having finite section rank. If $G$ does not include proper contranormal subgroups\, then $G$ is hypercentral group\, having hypercentral length at most $\\omega + k$ for some positive integer $k$ (here $\\omega$ is a first infinite ord inal). Moreover\, every its factor-group $G/H$ such that $\\Pi(G/H)$ is finite\, is nilpotent.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Francesca Dalla Volta (University of Milan-Bicocca) DTSTART:20210326T144000Z DTEND:20210326T151000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/35 DESCRIPTION:Title: Some result about Möbius functions for a finite non-solvable gro up\nby Francesca Dalla Volta (University of Milan-Bicocca) as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference) \n\n\nAbstract\n(Joint work with Giovanni Zini\, Università degli Studi d ella Campania ''Luigi Vanvitelli'')\n\n1. if $cL (G)$ is the subgroup la ttice of $G$. \nthe Möbius function $\\mu $ of $G$ is defined as $\\mu:\ \cL (G)~\\to~\\mathbb{Z}$\,\n$ H\\mapsto\\mu_{G}(H\,G) (= \\mu(H)$)\;\n\n2 . if $cC (G)$ is the poset of conjugacy classes of subgroups of $G$\, whe re $[H]\\leq[K]$ if and only if $H\\leq K^g$ for some $g\\leq G$\; its Mö bius function $cC\\to\\mathbb{Z}$\, $[H]\\mapsto\\mu_{cC}([H]\,[G]) = \\la mbda(H)$\n\n\nHawkes\, Isaacs and Özaydin \\cite{HIO} showed that \n$$\\m u(\\{1\\})=|G^\\prime|\\cdot\\lambda(\\{1\\})$$\nholds for any finite solv able group $G$\, and later Pahlings \\cite{Pahlings} proved more generally that the relation\n$$\n\\mu(H)=[N_{G^{\\prime}}(H):G^{\\prime} \\cap H]\\ cdot\\lambda(H)\n$$is true for any $H\\leq G$ whenever $G$ is finite and solvable.\n\n\nWe say that $G$ satisfies the $(\\mu\,\\lambda)$-property i f Equation \\eqref{eq:mulambda} holds for any $H\\leq G$.\n\n\nThe $(\\mu\ ,\\lambda)$-property does not hold for every finite group\; for instance\, it does not hold for the Mathieu group $M_{12}$ (see \\cite{Bianchi}) and for the unitary groups $U_3(2^{2^n})$ (see \\cite{Zini}).\n\nHere\, we pr esent some result about the relation between $\\mu$ and $\\lambda$ for so me classes of non-solvable groups\; among them\, the minimal non-solvable groups.\n\n\n\\begin{thebibliography}{99}\n\n\\bibitem{Bianchi} M. Bianch i\, A. Gillio Berta Mauri and A. Verardi\, On Hawkes-Isaacs-\\"Ozaydin's c onjecture\, \\textit{Rend. Ist. Lomb. Sc. e Lett.} {\\bf 124} (1990)\, 99- -117.\n\n\n\\bibitem{HIO} T. Hawkes\, M. Isaacs\, and M. \\"Ozaydin\, On t he M\\"obius function of a finte group\, \\textit{Rocky Mountain J. Math.} {\\bf 19} (4) (1989)\, 1003--1034.\n\\bibitem{Lucchini} A. Lucchini\, On the subgroups with non-trivial M\\"obius number\, \\textit{J. Group Theory } {\\bf 13} (2010)\, 589--600.\n\n\n\\bibitem{Mann1} A. Mann\, Positively finitely generated groups\, \\textit{Forum Math.} {\\bf 8} (4) (1996)\, 42 9--459.\n\n\\bibitem{Pahlings} H. Pahlings\, On the M\\"obius function of a finte group\, \\textit{Arch. Math. (Basel)} {\\bf 60} (1) (1993)\, 7--14 .\n\n\\bibitem{Zini} G. Zini\, The M\\"obius function of ${\\rm PSU}(3\,2^ {2^n})$\, \\textit{Ars Math. Comtemp.} {\\bf 16} (2) (2019)\, 377--401.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Organisers DTSTART:20210326T111000Z DTEND:20210326T112500Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/36 DESCRIPTION:Title: Remembering Karl Strambach\nby Organisers as part of Ischia G roup Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\nAbst ract: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Benjamin Klopsch (Heinrich-Heine-Universität Düsseldorf) DTSTART:20210326T121000Z DTEND:20210326T125000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/37 DESCRIPTION:Title: Automorphisms and pro-isomorphic zeta functions of class-$2$ nilp otent groups\nby Benjamin Klopsch (Heinrich-Heine-Universität Düssel dorf) as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24 -hour conference)\n\n\nAbstract\n(Joint work with M.\\\,N.~Berman and U.~O nn.)\n\n Pro-isomorphic zeta functions are one of the tools to study\n-- f rom an arithmetic point of view -- the distribution of\nfinite-index subgr oups in finitely generated nilpotent groups: they\nare Dirichlet generatin g series enumerating finite-index subgroups\nwhose profinite completion is isomorphic to that of the ambient group.\nWe study pro-isomorphic zeta fu nctions of $\\mathbb{Q}$-indecomposable\n$D^*$-groups. Loosely speaking\, these groups can be regarded as the\nbuilding blocks of finitely generate d class-two nilpotent groups with\nrank-two centre\, up to commensurabilit y\; a precise statement was given by\nGrunewald and Segal.\n\nSome time ag o we looked at Grunewald--Segal representatives of\n$\\mathbb{Q}$-indecomp osable $D^*$-groups of \\emph{odd} Hirsch length\n(Math.\\ Z.\\ 290\, 2018 ). In more recent work we studied\nGrunewald--Segal representatives of \\ emph{even} Hirsch length\, which\nhappen to form a richer and more varied class. For these groups\, we\ngive a complete description of the automorp hism groups of associated\ngraded Lie lattices. In particular\, this prov ides a key step towards\nthe elucidation of the pro-isomorphic zeta functi ons of such groups.\nUtilising our description of the automorphism groups\ , we calculate\nexplicitly the local pro-isomorphic zeta functions of spec ific\nexamples\; the resulting zeta functions are uniform and satisfy\nfun ctional equations.\n\nIn my talk I will try to explain the set-up\, the te rminology and some of the results\nhinted at above\, and I will aim to con vince you that there are still\nmany interesting open questions in the sub ject.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Primož Moravec (University of the Basque Country/Euskal Herriko U nibertsitatea) DTSTART:20210326T103000Z DTEND:20210326T110000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/38 DESCRIPTION:Title: Bogomolov multipliers of groups\nby Primož Moravec (Universi ty of the Basque Country/Euskal Herriko Unibertsitatea) as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\n\nA bstract\nBogomolov multipliers of groups are obstructions to Noether's pro blem in invariant theory of groups. On the other hand\, they appear in var ious different parts of mathematics\, such as K-theory\, representation th eory\, moduli theory and so on. We present an overview of some application s of these invariants\, as well as some recent developments.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Marino Cogliani DTSTART:20210326T130000Z DTEND:20210326T131000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/39 DESCRIPTION:Title: Intermezzo: Neapolitan Songs\nby Marino Cogliani as part of I schia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\ n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/39/ END:VEVENT BEGIN:VEVENT SUMMARY:Gunnar Traustason (University of Bath) DTSTART:20210326T131000Z DTEND:20210326T134000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/40 DESCRIPTION:Title: Powerfully nilpotent groups\nby Gunnar Traustason (University of Bath) as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC ( a 24-hour conference)\n\n\nAbstract\nIn this talk we discuss a special sub class of powerful groups called powerfully nilpotent groups. These are fin ite $p$-groups that possess a central series of a special kind. We will de scribe some structure theory and a `classification' in terms of an ancestr y tree and powerful coclass. We will also talk about some related subclass es of\npowerful groups.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Massimiliano Sala (University of Trento) DTSTART:20210326T135000Z DTEND:20210326T142000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/41 DESCRIPTION:Title: The group structure of elliptic curves over $\\mathbb{Z} /N \\mat hbb{Z}$\nby Massimiliano Sala (University of Trento) as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\n\n Abstract\n(Joint work with Daniele Taufer.)\n\nWe characterize the possibl e groups $E(Z/NZ)$ arising from elliptic curves over $\\mathbb{Z}/N \\math bb{Z}$ in terms of the groups $E(F_p)$\, with $p$ varying among the prime divisors of $N$. This classification is achieved by showing that the infin ity part of any elliptic curve over $\\mathbb{Z}/p^e \\mathbb{Z}$ is a $\\ mathbb{Z}/p^e \\mathbb{Z}$-torsor. As a first consequence\, when $E(\\math bb{Z}/N \\mathbb{Z})$ is a $p$-group\, we provide an explicit and sharp bo und on its rank. As a second consequence\, when $N=p^e$ is a prime power a nd the projected curve $E(F_p)$ has trace one\, we provide an isomorphism attack to the ECDLP\, which works only by means of finite rings arithmetic rather than involved methods.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Wolfgang Herfort (TU Wien) DTSTART:20210326T113000Z DTEND:20210326T120000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/42 DESCRIPTION:Title: Archipelago Groups are Locally Free\nby Wolfgang Herfort (TU Wien) as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24 -hour conference)\n\n\nAbstract\n(Joint work with Wolfram Hojka.)\n\nThe H armonic Archipelago (HA) is constructed as follows: In the plane consider circles of decreasing radius converging to a point of the plane. Then erec t cones of equal height above all these circles. \n\n\n During the talk a sketch of a proof that the fundamental group $\\pi_1(\\text{\\rm HA})$ i s locally free will be given. A consequence of this is a structure result for $H^1(\\text{\\rm HA})$.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrzej Żuk (Université de Paris VII (Denis Diderot)) DTSTART:20210325T194000Z DTEND:20210325T201000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/43 DESCRIPTION:Title: From PDEs to groups\nby Andrzej Żuk (Université de Paris VI I (Denis Diderot)) as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\n\nAbstract\nWe present a construction wh ich associates to differential equations discrete groups. In order to est ablish this relation we use automata and random walks on ultra discrete li mits.\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Organisers DTSTART:20210326T155500Z DTEND:20210326T160000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/44 DESCRIPTION:Title: Closing Ceremony\nby Organisers as part of Ischia Group Theor y\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\nAbstract: TBA\ n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/44/ END:VEVENT BEGIN:VEVENT SUMMARY:Organisers DTSTART:20210325T160000Z DTEND:20210325T161000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/45 DESCRIPTION:Title: Welcome Greetings\nby Organisers as part of Ischia Group Theo ry\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\nAbstract: TBA \n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/45/ END:VEVENT BEGIN:VEVENT SUMMARY:Organisers DTSTART:20210326T001000Z DTEND:20210326T002000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/52 DESCRIPTION:Title: A virtual tour of Procida\nby Organisers as part of Ischia Gr oup Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\nAbstr act: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/52/ END:VEVENT BEGIN:VEVENT SUMMARY:Organisers DTSTART:20210326T142500Z DTEND:20210326T143500Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/53 DESCRIPTION:Title: A virtual tour of Procida\nby Organisers as part of Ischia Gr oup Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conference)\n\nAbstr act: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/53/ END:VEVENT BEGIN:VEVENT SUMMARY:Organisers DTSTART:20210326T110500Z DTEND:20210326T111000Z DTSTAMP:20260417T030618Z UID:24HoursOfGOThIC/54 DESCRIPTION:Title: A virtual tour of La Mortella gardens\nby Organisers as part of Ischia Group Theory\, a.k.a. The 24 Hours of GOThIC (a 24-hour conferen ce)\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/24HoursOfGOThIC/54/ END:VEVENT END:VCALENDAR