Unitary units of the group algebra of modular groups

Let [Formula: see text] be the group algebra of the modular group [Formula: see text] over a finite field [Formula: see text] of characteristic two. We calculate the order of the ∗-unitary subgroup of the group algebra [Formula: see text] and describe the structure of the ∗-unitary subgroup in the case when [Formula: see text].

Structure of the unitary subgroup of the group algebra 𝔽2n(QD)16

In this paper, we established the structure of unitary unit subgroup [Formula: see text] of the group algebra [Formula: see text], where [Formula: see text] is the Quasi-dihedral [D. S. Dummit and R. Foote, Abstract Algebra, 3rd edn. (Wiley, 2004), pp. 71–72] (Semi-Dihedral [B. Huppert, Endliche Gruppen (Springer, 1967), pp. 90–93]) group of order 16 and [Formula: see text] is any finite field of characteristic 2 with [Formula: see text] elements.

Free pairs of symmetric and unitary units in normal subgroups of a division ring

Let [Formula: see text] be the field of fractions of the group algebra [Formula: see text] of the Heisenberg group [Formula: see text], over the field [Formula: see text] of characteristic [Formula: see text]. We show that for some involutions of [Formula: see text] that are not induced from involutions of [Formula: see text], [Formula: see text] contains free symmetric and unitary pairs. We also give a general condition for a normal unitary subgroup of a division ring to contain a free group, and prove a generalization of Lewin’s Conjecture.

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 ∗.

ON THE STRUCTURE OF THE UNITARY SUBGROUP OF THE GROUP ALGEBRA 𝔽2qD2n

We discuss the structure of the unitary subgroup V*(𝔽2qD2n) of the group algebra 𝔽2qD2n, where D2n = 〈x, y | x2n-1 = y2 = 1, xy = yx2n-1-1〉 is the dihedral group of order 2n and 𝔽2q is any finite field of characteristic 2, with 2q elements. We will prove that [Formula: see text], see Theorem 3.1.

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].

FREE SYMMETRIC AND UNITARY PAIRS IN DIVISION RINGS WITH INVOLUTION

Let D be a division ring with an involution and characteristic different from 2. Then, up to a few exceptions, D contains a pair of symmetric elements freely generating a free subgroup of its multiplicative group provided that (a) it is finite-dimensional and the center has a finite sufficiently large transcendence degree over the prime field, or (b) the center is uncountable, but not algebraically closed in D. Under conditions (a), if the involution is of the first kind, it is also shown that the unitary subgroup of the multiplicative group of D contains a free subgroup, with one exception. The methods developed are also used to exhibit free subgroups in the multiplicative group of a finite-dimensional division ring provided the center has a sufficiently large transcendence degree over its prime field.