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Concept# Symmetric group

Summary

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm{S}_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm{S}_n is n!.
Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set.
The symmetric group is important to

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Following the work of B. Kulshammer, J. B. Olsson and G. R. Robinson on generalized blocks of the symmetric groups, we give a definition for the l-defect of characters of the symmetric group G(n), where l > 1 is an arbitrary integer. We prove that the l-defect is given by an analogue of the hook-length formula, and use it to prove, when n < l(2), an l-version of the McKay conjecture in G(n).

In an article of 2003, Kulshammer, Olsson, and Robinson defined l-blocks for the symmetric groups, where l is an arbitrary integer, and proved that they satisfy an analogue of the Nakayama Conjecture. Inspired by this work and the definitions of generalized blocks and sections given by the authors, we give in this article a definition of d-sections in the finite general linear group, and construct d-blocks of unipotent characters, where d is an arbitrary integer. We prove that they satisfy one direction of an analogue of the Nakayama Conjecture, and, in some cases, prove the other direction. We also prove that they satisfy an analogue of Brauer's Second Main Theorem.

In a paper of 2003, Kulshammer, Olsson and Robinson defined l-blocks for the symmetric groups, where l > 1 is an arbitrary integer. In this paper, we give a definition for the defect group of the principal l-block. We then check that, in the Abelian case, we have an analogue of one of Broue's conjectures.

2008