On recognition of direct powers of finite simple linear groups by spectrum

TL;DR

This paper proves that for certain large powers of the simple linear group L_n(2), the spectrum uniquely determines the group among all finite groups, confirming a specific problem from the Kourovka Notebook.

Contribution

It establishes that the spectrum uniquely identifies the k-th direct power of L_n(2) when n is a sufficiently large power of 2, for all positive integers k.

Findings

Spectrum uniquely determines the group for large n

Validates a problem from the Kourovka Notebook

Provides conditions on n for unique recognition

Abstract

The spectrum of a finite group is the set of its element orders. We give an affirmative answer to Problem 20.58(a) from the Kourovka Notebook proving that for every positive integer $k$, the $k$-th direct power of the simple linear group $L_{n}(2)$ is uniquely determined by its spectrum in the class of finite groups provided $n$ is a power of $2$ greater than or equal to $56k^2$.