Normalized unit groups and their conjugacy classes
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Normalized unit groups and their conjugacy classes S KAUR and M KHAN∗ Department of Mathematics, Indian Institute of Technology Ropar, Nangal Road, Rupnagar 140 001, India *Corresponding author. Email: [email protected]
MS received 8 May 2018; revised 15 March 2019; accepted 31 May 2019 Abstract. Let G = H × A be a finite 2-group, where H is a non-abelian group of order 8 and A is an elementary abelian 2-group. We obtain a normal complement of G in the normalized unit group V (F G) and in the unitary subgroup V∗ (F G) over the field F with 2 elements. Further, for a finite field F of characteristic 2, we derive class size of elements of V (F G). Moreover, we provide class representatives of V∗ (F H ). Keywords.
Group ring; unitary subgroup; unit group; conjugacy class.
2010 Mathematics Subject Classification.
20C05, 16U60, 13C05, 20E45.
1. Introduction Let F G be a group algebra of a finite p-group G over a field F of characteristic p. Let ω(F G) denote the set of elements of F G with augmentation 0. For a normal subgroup H of G, let (H ) denote the ideal of F G generated by the elements {h − 1 | 1 = h ∈ H }. Clearly, (H ) = ω(F H ) · F G. The set ⎫ ⎧ ⎬ ⎨ ag g ∈ U(F G) ag = 1 V (F G) = ⎭ ⎩ g∈G
g∈G
is called the normalized unit group of the group algebra F G. It is known that V (F G) = 1 + ω(F G). Finding the structure of this class of p-groups is a very interesting problem in modular group algebras. The natural group homomorphism G → G/H defined by the rule g → g H can be extended to F-algebra homomorphism from F G onto F(G/H ) with (H ) as kernel. Since F is a field of characteristic p and G is a finite p-group, ω(F G) and hence (H ) is a nilpotent ideal of F G. Therefore, V (F G) ∼ = V (F(G/H )). 1 + (H ) The anti-automorphism g → g −1 of G can be extended linearly to an involution ∗ of F G. An element v of V (F G) is said to be unitary if v ∗ = v −1 . The set of all such elements forms a subgroup of V (F G) and is called the unitary subgroup V∗ (F G). © Indian Academy of Sciences 0123456789().: V,-vol
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In [6], Dennis posed the question: for which group G and ring R there exists a normal subgroup N of U(RG) such that U(RG) = N G? If N exists, then finding it is also a difficult problem. Several results regarding this problem in modular group algebras can be found in [8,9,12]. The conjugacy classes of a non-abelian group influence the study of its structure effectively. In [4], Coleman initiated the study of conjugacy classes of V (F G). In this direction, firstly, Rao and Sandling [13] proved that p can never occur as the cardinality of any conjugacy class of V (F G). Further, the results regarding conjugacy classes of V (F G) can be found in [1,2]. Let v be an element of V (F G). Then the centralizer of v in F G, C F G (v) = F ⊕ Cω(F G) (v), where Cω(F G) (v) is the centralizer of v in ω(F G). Thus the length of conjugacy class Cv of v in V (F G) is |Cv | =
|F||G|−1 |V (F G)| = = |F||G|−(dim F (C F G (v))) . |C V (F
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