SYMMETRY GROUPS OF SOME HADAMARD MATRICES

Abstract

 Hadamard matrices are a special class of square matrices with entries 1 and -1 only. They have  many applications in Coding Theory, Physics, Chemistry and Neural networks. Therefore, this paper makes  an attempt to study Hadamard matrices and their connection with Group Theory. Especially, we concentrate  on the Symmetry groups of Standard Hadamard matrices 0 1 2 3 H H H H , , , and 4 H . It is shown that the  Symmetry group of the Standard Hadamard matrices H0 and H1 is the trivial group and that of H2 is  S . Since Symmetry group of the Standard Hadamard matrix Hn is  

isomorphic to the Permutation group 3 

isomorphic to the General linear group of n n ⋅ invertible matrices over the field ℤ2 and the order of the  General linear group GL n q ( , ) of n n ⋅ invertible matrices over a finite field F containing q elements  n −1is ( ) ( )( )( ) ( ) ∏ − = − − − − ⋯ , it is shown that the orders of the  n n n n n 1 2 1− = k 0 q q q q q q q q q q Symmetry groups of H3 and H4 are 168 and 20,160 respectively.