Vol. 17, No. 1, 1966

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Classes of definite group matrices

Robert Charles Thompson

Vol. 17 (1966), No. 1, 175–190

Two positive definite symmetric n × n matrices A, B with integer elements and determinant one are said to be congruent if there exists an integral C such that B = CACT (CT is the transpose of C). This is an equivalence relation. The number of equivalence classes, C-classes, is finite and is known for all n 16. Let G be a finite group of order n and let Y , Z be two positive definite symmetric group matrices for G with integral elements and determinant one. If an integral group matrix X for G exists such that Z = XY XT then Z, Y are said to be G-congruent. G congruence is an equivalence relation. In this paper the interlinking of the G-classes with the C-classes is determined for all groups of order n 13. The principal result is that the G-class number is two for certain groups of orders eight or twelve and is one for all other groups of order n 13.

Mathematical Subject Classification
Primary: 15.30
Received: 9 November 1964
Published: 1 April 1966
Robert Charles Thompson