# Accession Number:

## AD0627630

# Title:

## UNIMODULAR GROUP MATRICES WITH RATIONAL INTEGERS AS ELEMENTS,

# Descriptive Note:

# Corporate Author:

## CALIFORNIA UNIV SANTA BARBARA DEPT OF MATHEMATICS

# Personal Author(s):

# Report Date:

## 1963-08-20

# Pagination or Media Count:

## 9.0

# 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 PL 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 PL g and PR g, are denoted LG and RG. It is noted that matrices in LG and RG commute, and a matrix that commutes with any PR g is a member of LG. These facts are used in an inductive proof on an ordered, ascending chain of subgroups of G to obtain the theorem. Author

# Descriptors:

# Subject Categories:

- Theoretical Mathematics