Unitary and Symmetric Units of a Commutative Group Algebra

Let F be a field of characteristic two and G a finite abelian 2-group with an involutory automorphism η. If G = H × D with non-trivial subgroups H and D of G such that η inverts the elements of H (H without a direct factor of order 2) and fixes D element-wise, then the linear extension of η to the group algebra FG is called a nice involution. This determines the groups of unitary and symmetric normalized units of FG. We calculate the orders and the invariants of these subgroups.

Nilpotency of Some Lie Algebras Associated with p-Groups

Let L = L0 + L1 be a 2-graded Lie algebra over a commutative ring with unity in which 2 is invertible. Suppose that L0 is abelian and L is generated by finitely many homogeneous elements a1,...,ak such that every commutator in a1,...,ak is ad-nilpotent. We prove that L is nilpotent. This implies that any periodic residually finite 2ʹ-group G admitting an involutory automorphism ϕ with CG(ϕ) abelian is locally finite.

A transitive self-polar double-twenty of planes

We demonstrate the existence, in the 5-dimensional projective space over any field J in which 1 + 1 ≠ 0 and −1 is a square, of a non-degenerate double-twenty of planes (ℋ, K) with the property that there is a group of collineations which acts transitively on ℋ ∪ K while each element of the group either maps ℋ onto itself and K onto itself or else swaps ℋ with K. If there is an involutory automorphism of J which swaps the two square roots of −1, then (ℋ, K) is also self-polar (with respect to a unitary polarity). We describe some of the geometry (in both 5-dimensional and 3-dimensional space) associated with the double-twenty (ℋ, K) and its group.