On the modular isomorphism problem for groups of class $3$

Mima Stanojkovski (Max-Planck-Institut Leipzig)

21-Jan-2021, 16:00-17:00 (3 years ago)

Abstract: Let $G$ be a finite group and let $R$ be a commutative ring. In 1940, G. Higman asked whether the isomorphism type of $G$ is determined by its group ring $RG$. Although the Isomorphism Problem has generally a negative answer, the Modular Isomorphism Problem (MIP), for $G$ a $p$-group and $R$ a field of positive characteristic $p$, is still open. Examples of $p$-groups for which the (MIP) has a positive solution are abelian groups, groups of order dividing $2^9$ or $3^7$ or $p^5$, certain groups of maximal class, etc.

I will give an overview of the history of the (MIP) and will present recent joint work with Leo Margolis for groups of nilpotency class $3$. In particular, our results yield new families of groups of order $p^6$ and $p^7$ for which the (MIP) has a positive solution and a new invariant for certain $2$-generated groups of class $3$.

group theory

Audience: researchers in the topic


GOThIC - Ischia Online Group Theory Conference

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