Involutory matrices, modulo m

Abstract

Given the prime power factorization of a positive integer m, a method for calculating the number of all distinct n x n - involutory matrices (mod m) is derived. This is done by first developing a method for the construction and enumeration of involutory matrices (mod P^α), without duplication, for each prime power modulus P^α. Using these results, formulas for the number of distinct involutory matrices (mod P^α) of order n are given where p is an odd prime, p=2, α= 1 and α > 1.

The concept of a fixed group associated with an involutory matrix (mod P^α) is used to characterize such matrices. Involutory matrices (mod P^α) of order n are considered as linear transformations on a vector space of n-tuples to provide uncomplicated proofs for the basic results concerning involutory matrices over a finite field.