Accession Number : AD0627630

Title :   UNIMODULAR GROUP MATRICES WITH RATIONAL INTEGERS AS ELEMENTS,

Corporate Author : CALIFORNIA UNIV SANTA BARBARA DEPT OF MATHEMATICS

Personal Author(s) : Thompson,R. C.

Report Date : 20 AUG 1963

Pagination or Media Count : 9

Abstract : The following theorem is proved: For a finite solvable group G, A is a unimodular matrix of rational integers such that B = AA' is a group matrix for G. Then A = A sub 1 T where A sub 1 is a unimodular group matrix of rational integers for G and T is a generalized permutation matrix. The left regular representation of G is defined by the matrix equations: (gg sub 1, gg sub 2, gg sub n) = (g sub 1, g sub 2, ..., g sub n) P(L) (g) where g sub 1, g sub 2, ..., g sub n are ordered elements of G and g E G. The right regular representation is similarly defined. Their group rings, the set of all linear combinations of P(L) (g) and P(R) (g), are denoted L*(G) and R*(G). It is noted that matrices in L*(G) and R*(G) commute, and a matrix that commutes with any P(R) (g) is a member of L*(G). These facts are used in an inductive proof on an ordered, ascending chain of subgroups of G to obtain the theorem. (Author)

Descriptors :   (*MATRICES(MATHEMATICS), GROUPS(MATHEMATICS)), (*GROUPS(MATHEMATICS), MATRICES(MATHEMATICS)), (*RATIONAL, MATRICES(MATHEMATICS), PERMUTATIONS, THEOREMS

Subject Categories : Theoretical Mathematics

Distribution Statement : APPROVED FOR PUBLIC RELEASE