Actions of nilpotent groups on nilpotent groups

TL;DR

This paper investigates the structure of the first cohomology set for finite nilpotent groups acting on each other, revealing a decomposition aligned with Sylow subgroups, and applies this to fixed point results in group actions.

Contribution

It provides a decomposition of the first cohomology set for finite nilpotent groups acting on nilpotent groups, extending understanding of their cohomological and fixed point properties.

Findings

Decomposition of H^1(J,N) in terms of Sylow p-subgroups.

Conditions ensuring a group element fixes a point in an action.

Extension of primary decomposition to non-abelian cases.

Abstract

For finite nilpotent groups $J$ and $N$, suppose $J$ acts on $N$ via automorphisms. We exhibit a decomposition of the first cohomology set in terms of the first cohomologies of the Sylow $p$-subgroups of $J$ that mirrors the primary decomposition of $H^1(J,N)$ for abelian $N$. We then show that if $N \rtimes J$ acts on some non-empty set $\Omega$, where the action of $N$ is transitive and for each prime $p$ a Sylow $p$-subgroup of $J$ fixes an element of $\Omega$, then $J$ fixes an element of $\Omega$.