Essential dimension, stable cohomological dimension, and stable cohomology of finite Heisenberg groups

TL;DR

This paper compares essential dimension and stable cohomological dimension of finite groups, establishing bounds and calculating the stable cohomological dimension for finite Heisenberg groups as exactly two.

Contribution

It introduces bounds for stable cohomological dimension based on group structure and computes this dimension precisely for finite Heisenberg groups.

Findings

Stable cohomological dimension is bounded by the length of a normal series with cyclic quotients.

The bound is not sharp for finite Heisenberg groups.

The stable cohomological dimension of H_p is exactly two.

Abstract

We compare the notions of essential dimension and stable cohomological dimension of a finite group G, prove that the latter is bounded by the length of any normal series with cyclic quotients for G, and show that, however, this bound is not sharp by showing that the stable cohomological dimension of the finite Heisenberg groups H_p, p any prime, is equal to two.