On the order of unitary subgroup of the modular group algebra 𝔽2kD2N

Let 𝔽qD2N be the group algebra of D2N, the dihedral group of order 2N, over 𝔽q = GF (q). In this paper, we compute the order of the unitary subgroup of the group of units of 𝔽2kD2N with respect to the canonical involution ∗.

UNITS IN F2D2p

Let p be an odd prime, D2p be the dihedral group of order 2p, and F2 be the finite field with two elements. If * denotes the canonical involution of the group algebra F2D2p, then bicyclic units are unitary units. In this note, we investigate the structure of the group [Formula: see text], generated by the bicyclic units of the group algebra F2D2p. Further, we obtain the structure of the unit group [Formula: see text] and the unitary subgroup [Formula: see text], and we prove that both [Formula: see text] and [Formula: see text] are normal subgroups of [Formula: see text].

Involutions of RA Loops

Let L be an RA loop, that is, a loop whose loop ring over any coefficient ring R is an alternative, but not associative, ring. Let ℓ ⟼ ℓθ denote an involution on L and extend it linearly to the loop ring RL. An element α ∈ RL is symmetric if αθ = α and skew-symmetric if αθ = –α. In this paper, we show that there exists an involution making the symmetric elements of RL commute if and only if the characteristic of R is 2 or θ is the canonical involution on L, and an involution making the skew-symmetric elements of RL commute if and only if the characteristic of R is 2 or 4.

UNITARY UNITS IN GROUP ALGEBRAS AND FIBONACCI SEQUENCES

Let * denote the canonical involution of the group algebra KG induced by the map x ↦ x-1 for x ∈ G. In case K is a real extension of ℚ, we consider Cayley unitary elements built out of skew elements k = α(x - x-1) in KG such that 1 + k is invertible in KG, for α ∈ K and x ∈ G. The constructions involve an interesting sequence in the coefficients of (1 + k)-1 which is the Fibonacci sequence when α = 1.

FINITE GROUPS WITH MULTIPLICITY-FREE PERMUTATION CHARACTERS

Let G be a finite group and H a subgroup of G. The Hecke algebra ℋ(G,H) associated with G and H is defined by the endomorphism algebra End ℂ[G]((ℂH)G), where ℂH is the trivial ℂ[H]-module and (ℂH)G = ℂH⊗ℂ[H] ℂ[G]. As is well known, ℋ(G,H) is a semisimple ℂ-algebra and it is commutative if and only if (ℂH)G is multiplicity-free. In [6], by a ring theoretic method, it is shown that if the canonical involution of ℋ(G,H) is the identity then ℋ(G,H) is commutative and, if there exists an abelian subgroup A of G such that G = AH then ℋ(G,H) is commutative. In this paper, by a character theoretic method, we consider the commutativity of ℋ(G,H).