Symmetry Groups of Some Hadamard Matrices

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 H₀ ,H₁ ,H₂ ,H₃ and H₄. It is shown that the Symmetry group of the Standard Hadamard matrices H₀ and H₁ is the trivial group and that of H₂ is isomorphic to the Permutation group S₃ . Since Symmetry group of the Standard Hadamard matrix H_n is 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 is it is shown that the orders of the Symmetry groups of H₃ and H₄ are 168 and 20,160 respectively.