Unitary Subgroups of Commutative Group Algebras of the Characteristic Two
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UNITARY SUBGROUPS OF COMMUTATIVE GROUP ALGEBRAS OF THE CHARACTERISTIC TWO Z. Balogh1 and V. Laver2,3
UDC 512.552.7
Let F G be the group algebra of a finite 2-group G over a finite field F of characteristic two and let ~ be an involution that arises from G. The ~-unitary subgroup of F G is denoted by V~ (F G) and defined as the set of all normalized units u satisfying the property u~ = u−1 . We establish the order of V~ (F G) for all involutions ~ arising from G, where G is a finite cyclic 2-group, and show that all ~-unitary subgroups of F G are not isomorphic.
1. Introduction Let F G be the group algebra of a 2-group G over a finite field F of characteristic two. The set of all units of F G that are mapped onto 1 by a complementary mapping forms a group. This group is denoted by V (F G) and called a group of normalized units. The description of the structure of V (F G) is a central problem of the theory of group algebras. Hence, it was studied in numerous works. A survey of the groups of units of modular group algebras can be found in [3]. Let ~ be an involution on F G. An element u 2 V (F G) is called ~-unitary if u~ = u−1 . The set of all ~-unitary elements of V (F G) forms a subgroup of V (F G), which is denoted by V~ (F G). A unitary subgroup corresponding to the canonical involution (F is a linear extension of the involution on G that associates each element of G with the inverse element) plays an important role in the investigation of the structure of groups of units of group algebras [7, 9, 10, 19, 20, 22]. The problem of determination of the order of V⇤ (F G) becomes especially complicated if the characteristic of F is equal to two. This problem was investigated in several works. In [8, 9], Bovdi and Szak´acs determined the structure of ⇤-unitary subgroups of all Abelian p-groups of finite fields with characteristic p. In [13], Bovdi and Grishkov obtained invariants of ⌘-unitary and symmetric normalized units of F G, where F is a finite field of two elements, G is a finite Abelian 2-group, and ⌘ is an involutory involution. In the case where a group G is not Abelian, we know only some separate results. Thus, in [11], the order of V⇤ (F G), where F is a finite field with characteristic two, was found for a dihedral group, a quaternion group, and extraspecial 2-groups. The structures of V⇤ (F G), where F is a field of two elements and G is either a group of order 16 or a group of the maximal class, were determined in [4] and [5], respectively. All group algebras ⇤-unitary subgroups of which are normal in V (F G) were described in [10]. The structures of V⇤ (F Q8 ) and V⇤ (F D8 ), where Q8 is a quaternion group, D8 is a dihedral group of order eight, and F is a finite field with characteristic two, were described in [14, 15]. In the nonmodular case, the number of available results is quite small. The order of V⇤ (F2k D2N ), where D2N is a dihedral group of order 2N, was found in [21]. 1
United Arab Emirates University, Al Ain, UAE; e-mail: [email protected]. Uzhhorod National University, Uzhhorod, Ukraine
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