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Questions about the branch of algebra that deals with groups.

21 votes
0 answers
468 views

Is there a "direct" proof of the Galois symmetry on centre of group algebra?

Let $G$ be a finite group, and $n$ an integer coprime to $|G|$. Then we have the following map, which is clearly not a morphism of groups in general: $$g\mapsto g^n.$$ This induces a linear automorphi …
Chris H's user avatar
  • 1,949
14 votes
2 answers
584 views

What is known about the structure of finite groups admitting an automorphism where all eleme...

Let $G$ be a finite group admitting an automorphism $\sigma$ of prime order $p$. Define the norm map $N:G\rightarrow G$ with respect to $\sigma$ by $N(g)= g\sigma(g)\sigma^2(g)\dotsb\sigma^{p-1}(g)$. …
Chris H's user avatar
  • 1,949
11 votes
1 answer
1k views

Are groups determined by their morphisms from solvable groups?

$\newcommand{\Grp}{\mathrm{Grp}}$Consider the category of groups $\Grp$, and within it we have the solvable groups $S$. Then any group $G$ determines the functor from solvable groups: $$h_G:=\text{hom …
Chris H's user avatar
  • 1,949
5 votes
1 answer
332 views

Is there a higher categorical structure which models the (higher) conjugation actions of a g...

Let $G$ be a group, and consider the action of $G$ on itself by conjugation. If we think of $G$ as a one object category, then the conjugation action can be realised as automorphisms of this category, …
Chris H's user avatar
  • 1,949
4 votes
0 answers
134 views

Is a group determined by the number of ways its elements multiply to the identity under some...

Let $G$ be a group, and for each ordered $n$ tuple $(g_1,...g_n)$ of elements of $G$, consider the function $f_n$ that outputs the number of permutations $\sigma\in S_n$ for which $g_{\sigma(1)}g_{\si …
Chris H's user avatar
  • 1,949
4 votes
0 answers
97 views

When can the trace on cohomology be computed as the Euler characteristic of fixed points?

In this question all groups are finite, and all spaces are nice (eg, simplicial sets). Given a $G$ space $X$, which we assume has finitely many nonzero cohomology groups, we can compute the trace of a …
Chris H's user avatar
  • 1,949