On the Faithfulness of the Representations of the Extraspecial 2-Groups Em-1

We consider a family of representations of the braid groups B-n corresponding to a specific solution to the Yang-Baxter equation. The images of the pure braid group P-n, a normal subgroup of B-n, under these representations are extraspecial 2-groups and the images of the braid group B-n are extensions of extraspecial 2-groups. We determine conditions under which any representation of the extraspecial 2-group, E-m(-1), is faithful. We then show that the irreducible representations of E-m(-1), constructed by Franko, Rowell and Wang, are faithful if and only if m = 2k or m = 2k - 1 (k odd); where as it is not faithful if m = 2k - 1 (k even).