ISSN 2148-838Xhttps://doi.org/10.13069/jacodesmath.1111746 J. Algebra Comb. Discrete Appl. 9(2) • 115–122 Received: 18 December 2021 Accepted: 8 January 2022 Journal of Algebra Combinatorics Discrete Structures and Applications On the isomorphism of unitary subgroups of noncommutative group algebras∗ Research Article Zsolt Adam Balogh Abstract: Let FG be the group algebra of a finite p-group G over a field F of characteristic p. Let ~ be an involution of the group algebra FG which arises form the group basis G. The upper bound for the number of non-isomorphic ~-unitary subgroups is the number of conjugacy classes of the automorphism group G with all the elements of order two. The upper bound is not always reached in the case when G is an abelian group, but for non-abelian case the question is open. In this paper we present a non-abelian p-group G whose group algebra FG has sharply less number of non-isomorphic ~-unitary subgroups than the given upper bound. 2010 MSC: 16S34, 16U60 Keywords: Group ring, Group of units, Unitary subgroup 1. Introduction Let FG be the group algebra of the group G over a field F. Let ~ be an involution of the group algebra FG. We say that the algebra involution ~ arises from the group G when ~ is an antiautomorphism on G. This antiautomorphism of G may also be called involution (for more details see in [19]). In this case the algebra involution ~ is the linear extension of the group involution ~ defined on G. A group algebra is always an algebra with involution, because the canonical ∗-involution of FG (the linear extension of the involution on G which sends each element of G to its inverse) exists for every F and G. The canonical involution ∗ on FG is a simple example of an algebra involution that arises from the group basis G. Let V (FG) denote the normalized unit group of FG, that is, the subgroup of the unit group of FG containing all units with augmentation 1. An element u ∈ V (FG) is called ~-unitary if u−1 = u~. The set of all ~-unitary units of FG forms a subgroup of V (FG), which is called ~-unitary subgroup and is denoted by V~(FG). Interest in the unitary subgroups arose in algebraic topology and unitary K-theory ∗ This work was supported by UAEU Research Start-up Grant No. G00002968. Zsolt Adam Balogh; Department of Mathematical Sciences, United Arab Emirates University, United Arab Emi- rates (baloghzsa@gmail.com). 115 https://orcid.org/0000-0001-5908-3443 Z. A. Balogh / J. Algebra Comb. Discrete Appl. 9(2) (2022) 115–122 introduced by Novikov [20]. The ∗-unitary subgroup is an actively investigated subgroup and it plays an important role of studying the structure of V (FG) for more details we refere the reader to Bovdi’s paper [9]). Let L be a finite Galois extension of F with Galois group G, where F is a finite field of characteristic two. A relation between the self-dual normal basis of L over F and the ∗-unitary subgroup of FG was discovered by Serre [21]. It was shown in [2] that the ∗-unitary subgroup of a group algebra determines the group basis G when it is a finite abelian p-group and F is a finite field of characteristic p. The structure of the unitary subgroups was studied in several papers (see [3], [4], [5], [12], [14], [15], [16], [17] and [22]). Let F be a field of characteristic p and G a nonabelian locally finite p-group. The groups G when V∗(FG) is normal in V (FG) are listed in [12]. Bovdi and Szakács [10] described the structure of the group V∗(FG) when G is a finite abelian p-group and F is a finite field of characteristic p. They also constructed a basis for V∗(FG) in [11]. The order of the unitary subgroup V∗(FG) is determined for finite p-groups and finite fields of characteristic p, if p is an odd prime (see in [13]). The order of V∗(FG) when p = 2 is an open question. It was determined only for some group classes (see in [1], [8] and [13]). The structure of V∗(F2G), where G is a 2-group of maximal class of order 8 or 16 and F2 is the field of two elements has been established in [6]. Additionally, the structures of V∗(FQ8) and V∗(FD8) are established in [16] and [18] respectively, where F is a finite field of characteristic 2, Q8 is the quaternion group of order 8 and D8 is the dihedral group of order 8. In the case when f is a homomorphism of G into the multiplicative group of the commutative ring K all the groups G whose f-unitary subgroup coincides with the unit group of KG are established in [9]. In [8] the invariants of the ~-unitary subgroup of FG are presented, when G is a finite abelian p-group, F is a field of p elements (p is an odd prime) and ~ is an involutory automorphism of G. In [3] an upper bound for the non-isomorphic ~-unitary subgroups is given, when ~ arises from G. The upper bound coincides the number of conjugacy classes of the automorphism group G with all the elements of order two including the identity map. In the case, when G is an abelian p-group the upper bound is not always sharp. A counterexample can be found in [4]. For non-abelian groups this question is open. In this paper we gave an example for a non-abelian p-group whose group algebra FG has less non-isomorphic ~-unitary subgroups than the given upper bound. 2. Involutions and unitary subgroups Let F be a finite field and G is either the dihedral group of order 8 or the quaternion group of order 8. In this section we show that the number of non-isomorphic ~-unitary subgroups of FG with respect to the involutions which arise from G is equals to the upper bound mentioned in the introduction. Let Aut G{2} be the set of all automorphism of G with the identity map. The composition of two antiautomorphisms ~ and ∗ of the group G is an automorphism of order two. Therefore, ~ can be considered as a composition of an automorphism of order two and the canonical involution, that is, ~ = φ◦∗, where φ ∈ Aut G{2}. We say that the involutions ~1 = φ1 ◦∗ and ~2 = φ2 ◦∗ are similar if φ1 is conjugate to φ2 in Aut G. We need the following lemma. Lemma 2.1. [3, Proposition 7] Let G be a group and F a field and let ~1 and ~2 be involutions of FG which arise from G. If ~1 is similar to ~2, then V~1 (FG) ∼= V~2 (FG). Let G be a finite group and let Λ2 denote the number of all distinct conjugacy classes of Aut G{2}. As a consequence of the previous lemma we have the following corollary. Corollary 2.2. [3, Corollary 8] Let G be a finite group and F a field. The number of non-isomorphic unitary subgroups of V (FG) with respect to the involutions which arise from G is at most Λ2. 116 Z. A. Balogh / J. Algebra Comb. Discrete Appl. 9(2) (2022) 115–122 In this section we show that the upper bound Λ2 is sharp for all the non-abelian groups of order 8. Moreover, we establish the structure of all non-isomorphic ~-unitary subgroups for these groups. First, let us consider the dihedral group D8 of order 8. It is well known that D8 ∼= Aut D8 and Aut G{2} is the union of four distinct conjugacy classes, that is, Λ2 = 4. Throughout this section we will use Lemma 2.4 in [1] free. Lemma 2.3. The number of non-isomorphic unitary subgroups of FD8 with respect to the involutions which arise from D8 is equals to Λ2, where |F| = 2n ≥ 2. Proof. It was shown in [18] that V∗(FD8) ∼= C25n o C2n. According to Lemma 2.1 it is enough to establish the structure of V~(FD8) when the involution ~ links to different conjugacy classes in Aut G{2}. Then Cσ1 = {σ1}, Cσ2, Cσ3 and Cσ4 are the distinct conjugacy classes of Aut G{2}, where σ1 is the identity map and σ2 : { a 7→ a3 b 7→ ab σ3 : { a 7→ a3 b 7→ a2b σ4 : { a 7→ a b 7→ a2b . Case σ2. Let α = ∑3 i=0 a i(αi + βib) ∈ FD8 where αi,βj ∈ F. Then α is ~-unitary if and only if αα~ = 1. A straightforward computation shows that αα~ equals to (α0 + α2) 2+(β1 + β3) 2a + (α1 + α3) 2a2+ (β0 + β2) 2a3 + δ1(1 + a)b + δ2(a 2 + a3)b, where δ1 = α0(β0 + β1) + α1(β1 + β2) + α2(β2 + β3) + α3(β0 + β3) and δ2 = α0(β2 + β3) + α1(β3 + β0) + α2(β0 + β1) + α3(β1 + β2). Clearly αα~ = 1 if and only if α0 + α2 = 1, β0 = β2, α1 = α3 and β1 = β3. Therefore δ1 = δ2 = β0 + β1 = 0, that is, β0 = β1 and every ~-unitary element can be written as α0 + α1a + (1 + α0)a 2 + α1a 3 + β0b + β0ab + β0a 2b + β0a 3b. Therefore V~(FD8) ∼= C23n. Case σ3. Let α = ∑3 i=0 a i(αi + βib) ∈ FD8, where αi,βj ∈ F. Then α~α = (α0 + α2 + β1 + β3) 2 + (α1 + α3 + β0 + β2) 2a2 + δ(1 + a2)b, where δ = (α0 + α2)(β0 + β2) + (α1 + α3)(β1 + β3). Clearly α~α = 1 if and only if α0 + α2 + β1 + β3 = 1, β0 = β2 and α1 = α3. Therefore every element of V~(FD8) is central or it can be written in the form either ab + x1 or a3b + x2, where x1,x2 ∈ ζ(V (FD8)). Since the exponent of ζ(V (FD8)) is two we have proved that V~(FD8) ∼= C25n. Case σ4. Let α = ∑3 i=0 a i(αi + βib) ∈ FD8, where αi,βj ∈ F. Then αα~ = (α0 + α1 + α2 + α3) 2 + (δ0 + δ1)(a + a 3) + (β0 + β1 + β2 + β3) 2a2 + (δ2 + δ3)(1 + a 2)b + (δ4 + δ5)(1 + a 2)ab, where δ0 = (α0 + α2)(α1 + α3), δ1 = (β0 + β2)(β1 + β3), δ2 = (α0 + α2)(β0 + β2), δ3 = (α1 + α3)(β1 + β3), δ4 = (α0 + α2)(β1 + β3), δ5 = (α1 + α3)(β0 + β2). Therefore, α0 + α1 + α2 + α3 = 1, β0 + β1 + β2 + β3 = 0, δ0 + δ1 = 0, δ2 + δ3 = 0 and δ4 + δ5 = 0. Since β1 + β3 = β0 + β2 we conclude that δ4 = δ2 and δ5 = δ3. Moreover, 0 = δ2 + δ3 = (α0 + α1 + α2 + 117 Z. A. Balogh / J. Algebra Comb. Discrete Appl. 9(2) (2022) 115–122 α3)(β0 + β2) = β0 + β2 and we have that β0 = β2, β1 = β3, δ1 = 0 and δ0 = 0. Thus, every ~-unitary element can be written as either a3 + α0Ĉ + α1Ĉa + β0Ĉb + β1Ĉab, if α2 = α0, or a2 + α0Ĉ + α1Ĉa + β0Ĉb + β1Ĉab, if α2 = 1 + α0. Let us denote by N the central elementary abelian subgroup 〈 1+α0Ĉ+α1Ĉa+β0Ĉb+β1Ĉab | αi,βi ∈ F 〉. Evidently, a2 ∈ N. Since a,a3 belong to the ~-unitary subgroup we have proved that V~(FD8) ∼= C4 ×C24n−1. It is well-known that Aut Q8 ∼= S4, where S4 is the symmetric group of order 24. It follows that Λ2 = 3. Lemma 2.4. The number of non-isomorphic unitary subgroups of FQ8 with respect to the involutions which arise from D8 equals Λ2, where |F| = 2n ≥ 2. Proof. Let σ1 be the identity automorphism of Q8. A straightforward computation shows that Aut G{2} = Cσ1 ∪Cσ2 ∪Cσ3, where σ2 : { a 7→ b b 7→ a σ3 : { a 7→ a3 b 7→ b . It was shown in [16] that V~(FQ8) ∼= Q8 ×C4n−12 . Let us consider the following two cases. Case i = 2. Let α = ∑3 i=0 a i(αi + βib) ∈ FQ8, where αi,βj ∈ F. Then α~α = (α0 + α2) 2 + δ1a + (β0 + β2) 2a2 + δ2a 3 + (β1 + β3) 2b + δ1ab+ (α1 + α3) 2a2b + δ2a 3b, where δ1 = α0(α1 + β1) + α2(α3 + β3) + β0(α1 + β3) + β2(α3 + β1), δ2 = α0(α3 + β3) + α2(α1 + β1) + β0(α3 + β1) + β2(α1 + β3). Evidently, α~α = 1 if and only if α0 + α2 = 1, β0 = β2, α1 = α3 and β1 = β3. They imply that δ1 = δ2 = α1 + β1 = 0, and so α1 = β1. Therefore every ~-unitary element can be written as a2 + α0Ĉa 2 + α1Ĉa + β0Ĉb + α1Ĉab. Thus V~(FQ8) ∼= C23n. Case i = 3. Let α = ∑3 i=0 a i(αi + βib) ∈ FQ8, where αi,βj ∈ F. Then αα~ = (α0 + α2 + β0 + β2) 2 + (α1 + α3 + β1 + β3) 2a2 + δ(1 + a2)b, where δ = (α0 + α2)(β0 + β2) + (α1 + α3)(β1 + β3). Let S~ = {αα~ |α ∈ V (FQ8)}. Clearly, S~ is a subgroup of ζ(V (FQ8)), therefore ψ : V (FQ8) → S~ (given by x 7→ xx~) is a homomorphism with kernel V~(FQ8). Thus |V~(FQ8)| = |V (FQ8)| |S~| = 27n 22n = 25n. 118 Z. A. Balogh / J. Algebra Comb. Discrete Appl. 9(2) (2022) 115–122 Let n = 1 and G~ = {g ∈ G |g~ = g−1}. It is easy to see that G~ = 〈 b 〉 and V~(FQ8) is a subgroup of G~ ·N, where N is an elementary abelian group. Since G~ ∼= C4, we get that V~(FQ8) ∼= C4 ×C32. Suppose that n > 1 and let ω1 and ω2 be elements of the unit group of F satisfying that ω1 6= 1 and ω1 + ω2 = 1. It is easy to see that b and ω1 + a + ω2b + ab are elements of V~(FQ8), but they are not commute. Therefore V~(FQ8) is not an abelian group. According to Theorem 2 in [7], the exponent of V (FQ8) is 4. Since b is a ~-unitary element with exponent 4 it follows that the exponent of V~(FQ8) is 4. Since |ζ(V (FQ8))| = 24n and x2 ∈ ζ(V (FQ8)) for all x ∈ V (FQ8) we have proved that V~(FQ8)/ζ(V (FQ8)) ∼= Cn2 . Therefore V~(FQ8) is a central extension of Cn2 by C2 4n. 3. Isomorphic unitary subgroups of noncommutative group alge- bra with non similar involutions In this section we present a non-abelian group whose group algebra has sharply less number of non-isomorphic ~-unitary subgroups than the given upper bound given in Corollary 2.2. Let H16 = 〈 a,c | a4 = b2 = c2 = 1, (a,b) = 1, (a,c) = b, (b,c) = 1 〉 be and let F be a finite field with |F| = 2n. The automorphism group of H16 is isomorphic to the following group 〈 σ1,σ2,σ3 | σ21 = σ 2 2 = σ 2 3 = σ 2 4 = σ 2 5 = 1, (σ1,σ2) = σ4, (σ1,σ3) = 1, (σ2,σ3) = σ5 〉, where σ1 =:   a 7→ a b 7→ a2b c 7→ c σ2 :   a 7→ a b 7→ bc c 7→ c σ3 :   a 7→ ab b 7→ b c 7→ c. Let us consider the following two automorphisms of order two in Aut H16 τ1 = σ1σ2σ5 :   a 7→ ac b 7→ a2bc c 7→ c and τ2 = (σ1,σ2) :   a 7→ a3c b 7→ bc c 7→ c. The conjugacy class of τ1 is Cτ1 = {σ1σ2σ5, σ1σ2σ4} and τ2 is a central element of the automorphism group. Theorem 3.1. Let ~1 = τ1 ◦ ∗ and ~2 = τ2 ◦ ∗ be involutions of H16 and let F be a finite field with |F | = 2n (n ≥ 1). Then ~1 is not similar to ~2 and V~1 (FH16) ∼= V~2 (FH16). Proof. First, we establish the structure of V~1 (FH16). Since every element of FH16 can be written as x =α0 + α1a + α2a 2 + α3a 3 + α4b + α5ab + α6a 2b + α7a 3b+ (α8 + α9a + α10a 2 + α11a 3 + α12b + α13ab + α14a 2b + α15a 3b)c (1) we have xx~ =(α0 + α2 + α8 + α10) 2 + (α5 + α7 + α13 + α15) 2a2 + δ1(a + a 3c)+ δ2(a 3 + ac) + δ3(b + a 2bc) + δ4(ab + abc) + δ5(a 2b + bc) + δ6(a 3b + a3bc)+ (α1 + α3 + α9 + α11) 2c + (α4 + α6 + α12 + α14) 2a2c, 119 Z. A. Balogh / J. Algebra Comb. Discrete Appl. 9(2) (2022) 115–122 where δ1 = (α0 + α10)(α1 + α11) + (α2 + α8)(α3 + α9) + (α4 + α14)(α5 + α15) + (α6 + α12)(α7 + α13) δ2 = (α0 + α10)(α3 + α9) + (α2 + α8)(α1 + α11) + (α4 + α14)(α7 + α13) + (α6 + α12)(α5 + α15) δ3 = (α0 + α10)(α4 + α14) + (α1 + α11)(α5 + α15) + (α2 + α8)(α6 + α12) + (α3 + α9)(α7 + α13) δ4 = (α0 + α8)(α5 + α13) + (α2 + α10)(α7 + α15) + (α4 + α12)(α3 + α11) + (α6 + α14)(α1 + α9) δ5 = (α0 + α10)(α6 + α12) + (α1 + α11)(α7 + α13) + (α3 + α9)(α5 + α15) + (α4 + α14)(α2 + α8) δ6 = (α0 + α8)(α7 + α15) + (α2 + α10)(α5 + α13) + (α1 + α9)(α4 + α12) + (α3 + α11)(α6 + α14). Evidently, x belongs to V~1 (FH16) if and only if xx ~1 = 1. Therefore α0 + α2 + α8 + α10 = 1, α5 + α7 + α13 + α15 = 0, α1 + α3 + α9 + α11 = 0, α4 + α6 + α12 + α14 = 0 and δ1 = δ2 = δ3 = δ4 = δ5 = δ6 = 0. Since α2 + α8 = 1 + α0 + α10 and α4 + α14 = α6 + α12 we have that δ1 = (α3 + α9) + (α0 + α10)(α1 + α11 + α3 + α9) + (α4 + α14)(α5 + α15 + α7 + α13) = α3 + α9, δ2 = (α1 + α11) + (α0 + α10)(α3 + α9 + α1 + α11) + (α4 + α14)(α7 + α13 + α5 + α15) = α1 + α11, δ3 = (α6 + α12) + (α0 + α10)(α4 + α14 + α6 + α12) + (α1 + α11)(α5 + α15 + α7 + α13) = α6 + α12, δ4 = (α7 + α15) + (α0 + α8)(α5 + α13 + α7 + α15) + (α4 + α12)(α3 + α11 + α1 + α9) = α7 + α15, δ5 = (α4 + α14) + α0 + α10)(α6 + α12 + α4 + α14) + (α1 + α11)(α7 + α13 + α5 + α15) = α4 + α14, δ6 = (α5 + α13) + (α0 + α8)(α7 + α15 + α5 + α13) + (α1 + α9)(α4 + α12 + α6 + α14) = α5 + α13. Therefore x = α0 + α2a 2 + α8c + α10a 2c + α1Ĉa + α4Ĉb + α5Ĉab, where Ĉ = 1 + a2 + c + a2c. As a consequence V~1 (FH16) is a central subgroup of V (FH16). Let N = 〈 1 + β1Ĉa, 1 + β2Ĉb, 1 + β3Ĉab | βi ∈ F 〉 be. Evidently, N ∼= C3n2 . Since a2Ĉ = cĈ = a2cĈ = Ĉ we conclude that N ∼= a2N ∼= cN ∼= a2cN. Since a2N · cN = a2cN and the pairwise intersections of N,a2N,a2cN are {1} we have proved that V~(FG) ∼= N ×a2N ×cN. Thus V~1 (FG) ∼= C9n2 . Now, we establish the structure of V~2 (FH16). Let x ∈ FH16 be. Using formula (1) we can compute the product xx~ = (α0 + α2 + α8 + α10) 2 + (α5 + α7 + α13 + α15) 2a2 + δ1(a + ac)+ δ2(a 3 + a3c) + δ3(b + bc) + δ4(ab + abc) + δ5(a 2b + a2bc) + δ6(a 3b + a3bc)+ (α4 + α6 + α12 + α14) 2c + (α1 + α3 + α9 + α11) 2a2c, where δ1 = (α0 + α8)(α9 + α1) + (α2 + α10)(α11 + α3) + (α4 + α12)(α5 + α13) + (α6 + α14)(α7 + α15), δ2 = (α0 + α8)(α11 + α3) + (α2 + α10)(α9 + α1) + (α4 + α12)(α7 + α15) + (α6 + α12)(α5 + α15), δ3 = (α0 + α8)(α12 + α4) + (α2 + α10)(α14 + α6) + (α1 + α9)(α15 + α7) + (α3 + α11)(α13 + α5), δ4 = (α0 + α8)(α13 + α5) + (α2 + α10)(α15 + α7) + (α4 + α12)(α1 + α9) + (α6 + α14)(α3 + α11), δ5 = (α0 + α8)(α14 + α6) + (α1 + α9)(α13 + α5) + (α2 + α10)(α12 + α4) + (α3 + α11)(α15 + α7), δ6 = (α0 + α8)(α15 + α7) + (α2 + α10)(α13 + α5) + (α1 + α9)(α14 + α6) + (α3 + α11)(α12 + α4). Keeping in mind that x belongs to V~2 (FH16), it follows that xx ~2 = 1. Therefore α0 + α2 + α8 + α10 = 1, α5 + α7 + α13 + α15 = 0, α1 + α3 + α9 + α11 = 0, α4 + α6 + α12 + α14 = 0 and δ1 = δ2 = δ3 = δ4 = δ5 = δ6 = 0. Since α0 + α8 = 1 + α2 + α10 and α4 + α12 = α6 + α14 we have that δ1 = α3 + α11, δ2 = α1 + α9, δ3 = α6 + α14, δ4 = α7 + α15, δ5 = α4 + α12 and δ6 = α5 + α13. Therefore α1 = α3 = α9 = α11, α6 = α4 = α12 = α14 and α5 = α7 = α13 = α15. 120 Z. A. Balogh / J. Algebra Comb. Discrete Appl. 9(2) (2022) 115–122 According to the above calculations we get that every x ∈ V~2 (FH16) can be written as x = α0 + α2a 2 + α8c + α10a 2c + α1Ĉa + α4Ĉb + α5Ĉab, where Ĉ = 1 + a2 + c + a2c, so V~2 (FH16) is a central subgroup of V (FH16). Let N = 〈 1+β1Ĉa, 1+β2Ĉb, 1+β3Ĉab | βi ∈ F 〉 be. Clearly, N ∼= C3n2 and N ∼= a2N ∼= cN ∼= a2cN because a2Ĉ = cĈ = a2cĈ = Ĉ. Since a2N · cN = a2cN and the pairwise intersections of N,a2N,a2cN are {1} we have proved that V~(FG) ∼= N ×a2N ×cN. Therefore we have V~1 (FG) ∼= V~2 (FG) ∼= C9n2 and the proof is completed. Corollary 3.2. 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