The third Milgram-Priddy class lifts

TL;DR

This paper proves that the third cohomology group of the finite general linear group over a finite field is non-zero and contains a unique element related to the Milgram-Priddy class.

Contribution

It establishes the non-triviality of the third cohomology of $GL_6(F_2)$ and links it to the Milgram-Priddy class, advancing understanding of group cohomology.

Findings

Third cohomology of $GL_6(F_2)$ is non-zero.

Identifies a unique non-trivial element in the cohomology.

Element restricts to the third Milgram-Priddy class.

Abstract

We show that the third cohomology of the finite general linear group $GL_6(\mathbb{F}_2)$ with trivial mod 2 coefficients is non-zero. The necessarily unique non-trivial element restricts to the third Milgram-Priddy class.