Rachel Quinlan

On the vanishing of subspaces of alternating bilinear forms

Given a field F and integer n≥3, we introduce an invariant sn (F) which is defined by examining the vanishing of subspaces of alternating bilinear forms on 2-dimensional subspaces of vector spaces. This invariant arises when we calculate the largest dimension of a subspace of n × n skew-symmetric matrices over F which contains no elements of rank 2. We show how to calculate sn (F) for various families of field F, including finite fields. We also prove the existence of large subgroups of the commutator subgroup of certain p-groups of class 2 which contain no non-identity commutators.

UNITARY GROUPS OVER LOCAL RINGS

Structural properties of unitary groups over local, not necessarily commutative, rings are developed, with applications to the computation of the orders of these groups (when finite) and to the degrees of the irreducible constituents of the Weil representation of a unitary group associated to a ramified extension of finite local rings.

Generic central extensions and projective representations of finite groups

Any free presentation for the finite group G determines a central extension (R, F ) for G having the projective lifting property for G over any field k. The irreducible representations of F which arise as lifts of irreducible projective representations of G are investigated by considering the structure of the group algebra kF , which is greatly influenced by the fact that the set of torsion elements of F is equal to its commutator subgroup and, in particular, is finite. A correspondence between projective equivalence classes of absolutely irreducible projective representations of G and F -orbits of absolutely irreducible characters of F ′ is established and employed in a discussion of realizability of projective representations over small fields.

On central difference sets in certain non-abelian 2-groups

In this note, we define the class of finite groups of Suzuki type, which are non-abelian groups of exponent 4 and class 2 with special properties. A group G of Suzuki type with |G| = 22s always possesses a non-trivial difference set. We show that if s is odd, G possesses a central difference set, whereas if s is even, G has no non-trivial central difference set.

Covering Groups of Rank 1 of Elementary Abelian Groups

ABSTRACT Covering groups of elementary Abelian groups of odd exponent p can be classified according to the rank of their pth power homomorphisms, which may be regarded as linear transformations of 𝔽 p –vector spaces. This article contains a description of the isomorphism types and the automorphism groups of those covering groups in which this rank is 1. Analogous considerations of elementary Abelian 2-groups and their covering groups are included in the final section.

Twisted group rings of metacyclic groups

Given a finite metacyclic group G, a central extension F having the projective lifting property over all fields is constructed. This extension and its group rings are used to investigate the faithful irreducible projective representations of G and the fields over which they can be realized. A full description of the finite metacyclic groups having central simple twisted group rings over fields is given.