On the decomposition matrices for the prime 2 of some simple groups
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On the decomposition matrices for the prime 2 of some simple groups

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Published by University of Birmingham in Birmingham .
Written in English


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Edition Notes

Thesis (Ph.D.) - University of Birmingham, Dept of Mathematics, 1984.

Statementby Thair Abdul Quadir Mohamad.
ID Numbers
Open LibraryOL13826885M

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The 2-modular decomposition matrices of the Mathieu group M 24 are in the directory M24, in the file The PDF file containing the 5-modular decomposition matrices of the McLaughlin group M c L and its triple cover 3.M c L is in the directory McL. ON REE'S SERIES OF SIMPLE GROUPS 63 Let P2 be a 2-Sylow subgroup of G. The modular characters for the prime 2 of LF(2,q) are as follows (facts the Cartan matrix must be [2] and the decomposition matrix m-Thus 9S restricted to 2-regular elements provides a modular character. Finally, since the only 2-regular class of defect 2 is that of. By the results of Granville and Ono [2], Michler [5], and Willems [9], whenever G is a finite non-Abelian simple group and p is a prime greater than 3, the group G has a p-block of defect zero. Special Matrix Definition Before further discussion, we first present definitions of some special matrices, here we follow the terms in [2]. Definition 1 A real matrix A is a symmetric matrix if it equals to its own transpose, that is A = AT. Definition 2 A complex matrix A is a hermitian matrix if it equals to its own complex.

  If -,~' consists of the two 2 X 2 matrices (0 01, (0 11 `1 0 0 0 ' then -/ possesses a primary decomposition; F' is the one and only primary component, and the 2 X 2 identity matrix is the one and only primary idempotent. EXAMPLE 2. If -/ consists of the two matrices (1 0), (0 0), 0 0 1 0 then the statements of the Theorem are not satisfied. No further re nements are possible since each subgroup in () has prime index (2 or 3) in the next subgroup. Our remaining examples involve matrix groups. Example For a eld F, the group A (F) has the normal series () ˆ 1 0 0 1 ˙ C ˆ 1 0 1 ˙ C ˆ 0 1 ˙; with factors isomorphic to Fand F. In Example, a normal series for S 4 was. Jordan introduced these three families as groups of matrices instead of groups of linear transforma-tions and proved they are simple when the field has prime order (except for a few trivial cases). 2. Range problem A different approach was taken by Otto Holder () when in he initiated what we will. Applicable to: m-by-n matrix A with linearly independent columns Decomposition: = where Q is a unitary matrix of size m-by-m, and R is an upper triangular matrix of size m-by-n Uniqueness: In general it is not unique, but if is of full rank, then there exists a single that has all positive diagonal elements. If is square, also is unique.; Comment: The QR decomposition provides an alternative.

has three 2-blocks, whose decomposition matrices are given in Tables 1{3, Tables 4{6, and Tables 7{9, respectively. We only print the rows of the decomposition matrices belonging to the irreducible ordinary characters being parametrized by 2-regular partitions, see Sections { The decomposition matrices . Prime graphs of simple groups & Goldbach conjecture. Indeed, recall that the order of a permutation is the ppcm of the orders of the cycles in its decomposition in a product of disjoint cycles ; and if 14 is the ppcm of some numbers, then 2 and 7 ; and in order to make an even permutation with at least one cycle of length 2 and at least one.   Let n ∈ N. The rows of the decomposition matrix of A n in characteristic 2 are distinct. M. Wildon / Journal of Algebra () – Proof. Suppose that the rows labelled by the ordinary characters χ and ψ are equal. Then χ(g)= ψ(g) for all 2 prime -elements of A n and so by Proposition , χ(g)= ψ(g) for all g ∈ A n. When ℓ = 0, the Decomposition Theorem of [BBD] implies that if A ∈ Perv L (N L, k) is simple, then I G L⊂P (A) is semisimple, so one can replace 'quotient' or 'subobject' in Definition