Commuting elements in central products of special unitary groups

We study the space of commuting elements in the central product Gm,p of m copies of the special unitary group SU(p), where p is a prime number. In particular, a computation for the number of path-connected components of these spaces is given and the geometry of the moduli space Rep(ℤn, Gm,p) of isomorphism classes of flat connections on principal Gm,p-bundles over the n-torus is completely described for all values of n, m and p.

Computing tensor decompositions of finite matrix groups

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience We describe an algorithm to compute tensor decompositions of central products of groups. The novelty over previous algorithms is that in the case of matrix groups that are both tensor decomposable and imprimitive, the new algorithm more often outputs the more desirable tensor decomposition.

Extra-Special Groups of Order 32 as Galois Groups

In this article we examine conditions for the appearance or nonappearance of the two extra-special 2-groups of order 32 as Galois groups over a field F of characteristic not 2. The groups in question are the central products DD of two dihedral groups of order 8, and DQ of a dihedral group with the quaternion group, obtained by identifying the central elements of order 2 in each factor group. It is shown that the realizability of each of these groups as Galois groups over F implies the realizability of other 2-groups (which are not their quotient groups), and in turn that realizability of certain other 2-groups implies the realizability of DD and DQ. We conclude by providing an explicit construction of field extensions with Galois group DD.

More on finite groups with few defining relations

Certain central products of the binary polyhedral groups with finite cyclic groups are here shown to have presentations with two generators and two defining relations; this disproves a conjecture of the second author, stated in J. Austral. Math. Soc. Ser. A 38 (1985), 230–240.