Local functions on blocks and automorphisms of partial groups

Mason, Jamie ORCID: 0000-0003-2162-0893 (2024). Local functions on blocks and automorphisms of partial groups. University of Birmingham. Ph.D.

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Abstract

In the first chapter of this thesis we define a block-by-block version of Isaacs and Navarro’s chain local condition and then prove that the Alperin–McKay conjecture is equivalent to a certain function on groups having this property. We then go on to prove several other block-by-block versions of results from Isaacs and Navarro’s paper. The results in this chapter have also been published in [35].

The second part concerns automorphisms of partial groups, specifically which groups can arise as automorphisms of different types of partial group. We show that for any finite group one can construct a finite partial group that has this finite group as an automorphism group. We also prove an analogous result for groups and objective partial groups as well as a partial result for finite groups and finite objective partial groups. Lastly we show that there are no automorphism groups of localities that do not arise as automorphism groups of groups, a rephrasing of the same result for fusion systems.

This thesis is split into two entirely self-contained chapters.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):