Universal Covers of Finite Groups
24th February 2021, 4:00 pm – 5:00 pm
The textbook treatment of group extensions is through the tool of group cohomology. While it provides a satisfying description in setting the set of extensions in bijection to a vector space, it makes it hard to understand how different extensions relate to each other, or to work out all but the smallest examples by hand.
When we consider extensions instead as larger quotients of the same free group, a theorem by Gaschuetz gives a beautiful theory that connects the possible extensions with the modular representation theory of the quotient group. This makes it possible to construct an universal cover (dependent on the factor group, the module, and the number of generators) of a finite group that generalizes the concept of the $p$-cover for $p$-groups. This in turn provides for a new algorithm for finding larger quotients of finitely presented groups. (Joint work with H.Dietrich, Melbourne)