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Perfect Isometry Groups for Blocks of Finite Groups

Ruengrot, Pornrat

[Thesis]. Manchester, UK: The University of Manchester; 2011.

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Abstract

Our aim is to investigate perfect isometry groups, which are invariants for blocks of finite groups. There are two subgoals. First is to study some properties of perfect isometry groups in general. We found that every perfect isometry has essentially a unique sign. This allowed us to show that, in many cases, a perfect isometry group contains a direct factor generated by -id.The second subgoal is to calculate perfect isometry groups for various blocks. Notable results include the perfect isometry groups for blocks with defect 1, abelian p-groups, extra special p-groups, and the principal 2-block of the Suzuki group Sz(q). In the case of blocks with defect 1, we also showed that every perfect isometry can be induced by a derived equivalence. With the help of a computer, we also calculated perfect isometry groups for some blocks of sporadic simple groups.Apart from perfect isometries, we also investigated self-isotypies in the special case where C_G(x) is a p-group whenever x is a p-element. We applied our result to calculate isotypies in cyclic p-groups and the principal 2-blocks of the Suzuki group Sz(q).

Bibliographic metadata

Type of resource:
Content type:
Form of thesis:
Type of submission:
Degree type:
Doctor of Philosophy
Degree programme:
PhD Mathematical Sciences
Publication date:
Location:
Manchester, UK
Total pages:
176
Abstract:
Our aim is to investigate perfect isometry groups, which are invariants for blocks of finite groups. There are two subgoals. First is to study some properties of perfect isometry groups in general. We found that every perfect isometry has essentially a unique sign. This allowed us to show that, in many cases, a perfect isometry group contains a direct factor generated by -id.The second subgoal is to calculate perfect isometry groups for various blocks. Notable results include the perfect isometry groups for blocks with defect 1, abelian p-groups, extra special p-groups, and the principal 2-block of the Suzuki group Sz(q). In the case of blocks with defect 1, we also showed that every perfect isometry can be induced by a derived equivalence. With the help of a computer, we also calculated perfect isometry groups for some blocks of sporadic simple groups.Apart from perfect isometries, we also investigated self-isotypies in the special case where C_G(x) is a p-group whenever x is a p-element. We applied our result to calculate isotypies in cyclic p-groups and the principal 2-blocks of the Suzuki group Sz(q).
Thesis main supervisor(s):
Thesis co-supervisor(s):
Thesis advisor(s):
Language:
en

Institutional metadata

University researcher(s):

Record metadata

Manchester eScholar ID:
uk-ac-man-scw:142297
Created by:
Ruengrot, Pornrat
Created:
21st December, 2011, 12:45:18
Last modified by:
Ruengrot, Pornrat
Last modified:
6th April, 2012, 18:24:12

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