The Centre of a Block

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Abstract

Let G be a finite group and F a field. The group algebra FG decomposes as a direct sum of two-sided ideals, called the blocks of FG. In this thesis the structure of the centre of a block for various groups is investigated. Studying these subalgebras yields information about the relationship between two block algebras and, in certain cases, forms a vital tool in establishing the non-existence of an important equivalence in the context of modular representation theory. In particular, the focus lies on determining the Loewy structure for the centre of a block, which so far has not been studied in detail but is fundamental in gaining a better understanding of the block itself.For finite groups G with non-abelian, trivial intersection Sylow p-subgroups, the analysis of the Loewy structure of the centre of a block allows us to deduce that a stable equivalence of Morita type does not induce an algebra isomorphism between the centre of the principal block of G and its Sylow normaliser. This was already known for the Suzuki groups; the techniques will be generalised to extend the result to cover the Ree groups of type ^2G_2(q).In addition, the three sporadic simple groups with the trivial intersection property, M_11, McL and J_4, together with some small projective special unitary groups are studied with respect to showing the non-existence of an isomorphism between the centre of the principal block and the centre of its Brauer correspondent.Finally, the Loewy structure of centres of various principal blocks are calculated. In particular, some small sporadic simple groups and groups with normal, elementary abelian Sylow p-subgroups are considered. For the latter, some specific formulae for the Loewy length are derived, which generalises recent results on groups with cyclic Sylow p-subgroups.

Keyword(s)

Representation Theory

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