Cosets of normal subgroups: what can they tell us about the group

Abstract: In this talk we discuss how information about the non-trivial cosets of a normal subgroup can influence the structure of a finite group.

Here is one example:

Let $G$ be a finite group with a normal subgroup $N$. Assume that for each non-trivial coset $Nr$ all the elements in $Nr$ have the same order.

Then one of the following holds:

1. $N < F(G)$, in this case $F(G)$, and hence $N$, is a $p$-group for some prime $p$.

2. $N = F(G)$.

3. $F(G) < N$ in this case $G/N$ is a $p$-group.

Here $F(G)$ is the Fitting subgroup of $G$.

Mathematics

Audience: researchers in the topic

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ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.

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