This preprint is translated from the original journal publication in Russian: A. Konovalov and A. Tsapok, Symmetric subgroups of the normalised unit group of the modular group algebra of a finite p-group, Nauk. Visn. Uzhgorod. Univ., Ser. Mat., 9 (2004), 20–24.Let V(KG) be a normalised unit group of the modular group algebra of a finite p-group G over the field K of p elements. We introduce a notion of symmetric subgroups in V(KG) as subgroups invariant under the action of the classical involution of the group algebra KG. We study properties of symmetric subgroups and construct a counterexample to the conjecture by V.Bovdi, which states that V(KG)=, where S* is a set of symmetric units of V(KG).PreprintPeer reviewe

Konovalov, Alexander

Krivokhata, A. G.

English

This preprint is translated from the original journal publication in Russian: A. Konovalov and A. Tsapok, Symmetric subgroups of the normalised unit group of the modular group algebra of a finite p-group, Nauk. Visn. Uzhgorod. Univ., Ser. Mat., 9 (2004), 20–24.Let V(KG) be a normalised unit group of the modular group algebra of a finite p-group G over the field K of p elements. We introduce a notion of symmetric subgroups in V(KG) as subgroups invariant under the action of the classical involution of the group algebra KG. We study properties of symmetric subgroups and construct a counterexample to the conjecture by V.Bovdi, which states that V(KG)=, where S* is a set of symmetric units of V(KG).Peer reviewe

St Andrews Research Repository

arXiv:0801.0809v1  [math.RA]  5 Jan 2008SYMMETRIC SUBGROUPS IN MODULAR GROUP ALGEBRASA.B. KONOVALOV, A.G. KRIVOKHATAAbstract. Let V (KG) be a normalised unit group of the modular groupalgebra of a finite p-group G over the field K of p elements. We introducea notion of symmetric subgroups in V (KG) as subgroups invariant under theaction of the classical involution of the group algebra KG. We study propertiesof symmetric subgroups and construct a counterexample to the conjecture byV. Bovdi, which states that V (KG) = 〈G, S∗〉, where S∗ is a set of symmetricunits of V (KG).1. IntroductionIn this preprint1 we investigate the structure of the unit group U(KG) of themodular group algebra KG, where G is a finite p-group and K is a field of pelements. U(KG) is a direct product of the unit group of the field K and thenormalized unit group V (KG) that consists of all elements of the form 1 + x,where x belongs to I(G), the latter being the augmentation ideal of the modulargroup algebra KG. Thus, the task of the investigation of U(KG) is reduced to theinvestigation of the p-group V (KG).Study of units and their properties is one of the main research problems in groupring theory. Results obtained in this direction are also useful for the investigationof Lie properties of group rings, isomorphism problem and other open questions inthis area (see, for example, [2]).Let us consider the mapping ϕ from KG into itself which maps an elementx =∑λgg to∑λgg−1. This mapping is an antiautomorphism of order two,and it is called the classical involution of the group algebra KG. In what follows,we will denote ϕ(x) by x∗. An element x ∈ KG will be called symmetric, if x = x∗.Note that if x∗ 6= x−1, then x∗x— nontrivial symmetric element. Indeed, ϕ(x∗x) =(x∗x)∗ = x∗x∗∗ = x∗x since x∗∗ = x. Classical involution and symmetric elementswere studied by V. Bovdi, M. Dokuchaev, L. Kovacs, S. Sehgal in [3, 4, 5, 6, 8].In [9] the generating system for V (KG) was given, however that system is mini-mal only in the abelian case. In the general case, the problem of the determinationof the minimal generating system of V (KG) remains open. For groups of smallorder the minimal generating system can be determined with the help of computerafter the construction of V (KG) using the algorithm from [9] implemented in theLAGUNA package [10] for the computational algebra system GAP [11]. Obtainingnew results about the minimal generating system would be useful for the improve-ment of algorithms for the computation of V (KG).1991 Mathematics Subject Classification. Primary 16S34, 20C05.Key words and phrases. modular group algebra, classical involution, normalized unit group,symmetric unit, symmetric subgroup, computational algebra system GAP.1This preprint is a slightly updated translation of the paper [7], earlier written by the sameauthors in Russian.12 A.B. KONOVALOV, A.G. KRIVOKHATAIn 1996 V. Bovdi suggested the following conjecture:Conjecture 1. Let G be a finite nonabelian p-group, V (KG) be the normalizedunit group of the modular group algebra KG, and S∗ = {x ∈ V (KG) | x∗ = x} bethe set of symmetric units. Is it true that(1) V (KG) = 〈G,S∗〉?In the present paper we give a counterexample to this conjecture and investigatefurther properties of the classical involution. We introduce the notions of symmetricsubsets of the group algebra and symmetric subgroups of V (KG), and investigatetheir properties. Then we give some obvious examples of symmetric subgroupsand formulate the question about the existence of non-trivial symmetric subgroup.After this, we consider the normalized unit group of the modular group algebraof the quaternion group of order 8, and check for it the conjecture (1). Using theLAGUNA package [10], we discovered that in this case the condition (1) does nothold, because 〈G,S∗〉 has the order 64. In the present paper we give a purelytheoretical proof of this fact. Besides this, it appears that 〈G,S∗〉 is the symmetricsubgroup, as well as the set S∗, which in this case is also a subgroup. Thus, weobtain two examples of non-trivial symmetric subgroups.It would be interesting to continue studies of properties of symmetric subgroupsof V (KG) and conditions of the existence of non-trivial symmetric subgroups inV (KG).2. Symmetric subgroupsLet H be a subset of the group algebra KG. Then H will be called symmetric,if H∗ = H , where H∗ = {h∗|h ∈ H}. Similarly, a subgroup H ⊆ V (KG) will becalled a symmetric subgroup, if H∗ = H .Clearly, symmetric subgroups exist. It is easy to see that {1}, G and V (KG)are trivial examples of symmetric subgroups. This naturally raises a question ofthe existence of modular group algebras that possesses non-trivial symmetric sub-groups. Moreover, in [3] conditions were obtained under which the set of symmetricelements forms a group, that in this case will be symmetric. Therefore, it would beinteresting to find an example of non-trivial symmetric subgroup that contains non-symmetric units, thus the restriction of the classical involution on this subgroupwill be not an identity mapping.Before we construct such example, we will state and prove some properties ofsymmetric subgroup.Lemma 1. If H is a subgroup of V (KG), then H∗ also is a subgroup of V (KG).Proof. We will check that H∗ is a subgroup of V (KG). Since 1 ∈ H , we have1 = ϕ(1) ∈ H∗. Then, if a∗, b∗ ∈ H∗, so does a∗b∗ ∈ H∗, since a∗b∗ = ϕ(ba), andba ∈ H . Finally, if a∗ ∈ H∗, then (a∗)−1 = (ϕ(a))−1 = ϕ(a−1) ∈ H∗, and thelemma is proved.Lemma 2. Let H ′ and H be subgroups of V (KG), and H be a symmetric subgroup.If H ′ is a conjugate of H with the help of an element g ∈ V (KG) such that (g∗)−1 =(g−1)∗, then H ′∗ is also a conjugate of H with the help of element (g∗)−1.Proof. Let us assume that g ∈ V (KG) is such that H ′ = g−1Hg, and H = H∗.Then H ′∗ = (g−1hg)∗ = g∗h(g−1)∗ = g∗H(g−1)∗. If (g∗)−1 = (g−1)∗, what holds,SYMMETRIC SUBGROUPS IN MODULAR GROUP ALGEBRAS 3in particular, for all g ∈ G, then H ′∗ = H(g−1)∗ , and the lemma is proved. Notethat H ′ is not necessary to be symmetric, and also that H ′∗ = (Hg−1)∗.Lemma 3. Let H1 and H2 be symmetric subgroups. Then H1H2 also is a sym-metric subgroup.Proof. Let H1 and H2 be symmetric subgroups. Consider an element fromH1H2 having the form h = a1b1a2b2 . . . akbk, where ai ∈ H1 and bi ∈ H2. Thenh∗ = ukvk . . . u2v2u1v1, where ui = b∗i , vi = a∗i , also belongs to H1H2, because H1and H2 are symmetric. Thus, the lemma is proved.3. Investigation of the conjecture on generators of V (KG)To verify the conjecture by V. Bovdi about generators of the normalized unit groupwe used the LAGUNA package [10] for the computational algebra system GAP[11]. We developed a program in GAP language using functions available in theLAGUNA package, that performed the following steps:a) get the set of all elements of the group G;b) compute the normalized unit group V (KG) (generated by group algebraelements) with the help of the LAGUNA package, and the group W thatis a power-commutator presentation of V (KG);c) construct the embedding f from G to the group algebra KG;d) construct the isomorphism ψ from V (KG) to the group W ;e) compute the set S of symmetric units of the group algebra KG;f) using the isomorphism ψ, find images of the set of symmetric units S andf(G) in the group W ;g) compute the subgroup 〈ψ(f(G)), ψ(S∗)〉 ⊆W and verify the condition (1).As a result, we find out that the condition (1) does not hold for the quaterniongroup of order eight.4. Computation of the symmetric subgroup 〈G,S∗〉In this section we will give a theoretical explanation why the quaternion group Q8provides a counterexample to the conjecture by V. Bovdi. The group Q8 is givenby the following presentation:Q8 = 〈a, b|a4 = b4 = 1, a2 = b2, b−1ab = a3〉,and consists of the following elements:Q8 = {aibj, 0 ≤ i ≤ 4, 0 ≤ j ≤ 2}.Every symmetric element x ∈ S∗ has the form(2) x = α0 + α1a2 + γx,where(3) γx = α2(a+ a3) + α3(b+ a2b) + α4(ab+ a3b).Since x is a unit from V (KQ8), we have that α0 +α1 = 1 (i.e. one and only one ofcoefficients α0, α1 is equal to one). From this follows that |S∗| = 16.Lemma 4. Let S∗ = {s ∈ V (KQ8)|s∗ = s} be the set of symmetric units. ThenS∗ ⊂ Z(KQ8), where Z(KQ8) is the center of the group algebra KQ8.4 A.B. KONOVALOV, A.G. KRIVOKHATAProof. The group Q8 has the following decomposition on the conjugacy classesof elements:Q8 = {1} ∪ {a2} ∪ {a, a3} ∪ {b, a2b} ∪ {ab, a3}.Since class sums are central elements (see [1]), the lemma follows immediately from(2) and (3).Lemma 5. The set S∗ is a subgroup of the normalized unit group V (KQ8).Proof. The lemma follows from the result from [3] stating that the set of sym-metric units is a subgroup if and only if they all commute, in combination with thelemma 4.It also follows from the lemma 5 that S∗ is a non-trivial symmetric subgroup ofV (KQ8). However, all its elements are symmetric, so we did not have an exampleof a symmetric subgroup which contain non-symmetric units yet.Lemma 6. ∀x ∈ S∗ x2 = 1.Proof. By (2), we have thatx2 = α20 + α1a4 + γ2x = α20 + α21 = 1,since the characteristic of the field K is two, elements 1, a4 and γx are in the centerZ(KQ8), and γ2x = 0.Now we are ready to prove the main statement, that gives a counterexample tothe conjecture by V. Bovdi.Theorem 1. Let Q8 be the quaternion group of order eight, V (KQ8) be the nor-malized unit group of its modular group algebra over the field of two elements, andS∗ = {x ∈ V (KQ8)|x∗ = x} be the set of symmetric units. Then|H | = |〈Q8, S∗〉| = 64.Proof. Since S∗ ⊂ Z(KQ8), then each element from H can be written as x = gs,where g ∈ Q8 and s ∈ S∗. Note that Q8⋂S∗ = {1, a2} that coincides with Z(Q8).Thus, 〈Q8, S∗〉 is a central product of groups Q8 and S∗. From this follows that|H | = |S∗|·|Q8||Z(Q8)| =16·82 = 64, and the theorem is proved.Thus, we proved that the order of a subgroup generated by the group Q8 andthe set of symmetric elements S∗ is equal to 64, so it does not coincide with thenormalized unit group V (KQ8) of order 128. So, H 6= V (KQ8), and the quaterniongroup of order eight gives a counterexample to the conjecture by V. Bovdi. Besidesthis, H = 〈Q8, S∗〉 is a non-trivial symmetric subgroup of the group V (KQ8),containing both symmetric and non-symmetric units.References[1] A.A. Bovdi, Group rings, UMK VO, Kiev, 1988.[2] A.A. Bovdi and J. Kurdics, Lie properties of the group algebra and the nilpotency class ofthe group of units, J. Algebra 212 (1999), no. 1, 28–64. MR1670626 (2000a:16051)[3] V. Bovdi, L.G. Kovacs and S.K. Sehgal, Symmetric units in modular group algebras, Comm.Algebra 24 (1996), no. 3, 803–808. MR1374635 (97a:16064)[4] V. Bovdi, On symmetric units in group algebras, Comm. Algebra 29 (2001), no. 12, 5411–5422. MR1872239 (2002j:16035)[5] V. Bovdi and M. Dokuchaev, Group algebras whose involutory units commute, Algebra Col-loq. 9 (2002), no. 1, 49–64. MR1883427 (2003a:16045)[6] V. Bovdi and T. Rozgonyi, On the unitary subgroup of modular group algebras, Acta Math.Acad. Paedagog. Nyha´zi., 13/D (1992), 13–17.SYMMETRIC SUBGROUPS IN MODULAR GROUP ALGEBRAS 5[7] A. Konovalov and A. Tsapok, Symmetric subgroups of the normalised unit group of themodular group algebra of a finite p-group, Nauk. Visn. Uzhgorod. Univ., Ser. Mat., 9 (2004),20–24.[8] S.K. Sehgal, Symmetric elements and identities in group algebras, in Algebra, 207–213,Birkha¨user, Basel. MR1690798 (2000f:16035)[9] A. Bovdi, The group of units of a group algebra of characteristic p, Publ. Math. Debrecen52 (1998), no. 1-2, 193–244. MR1603359 (99b:16051)[10] V. Bovdi, A. Konovalov, C. Schneider and R. Rossmanith, LAGUNA — Lie AlGebras andUNits of group Algebras, Version 3.2.1; 2003 (http://ukrgap.exponenta.ru/laguna.htm).[11] The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.3; 2002(http://www.gap-system.org).A.B. KonovalovSchool of Computer Science, University of St Andrews,Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, ScotlandE-mail address: konovalov@member.ams.orgA.G. Krivokhata (nee Tsapok)Department of Mathematics, Zaporozhye National University,66 Zhukovskogo str., 69063, Zaporozhye, UkraineE-mail address: k-algebra@zsu.zp.ua

Algorithms, and Programming, Version 4.3;

Group algebras whose involutory units commute,

Group rings,

LAGUNA — Lie AlGebras and UNits of group Algebras, Version 3.2.1;

Lie properties of the group algebra and the nilpotency class of the group of units,

On symmetric units in group algebras,

On the unitary subgroup of modular group algebras,

Symmetric elements and identities in group algebras,

Symmetric subgroups of the normalised unit group of the modular group algebra of a finite p-group,

Symmetric units in modular group algebras,

The group of units of a group algebra of characteristic p,

Symmetric subgroups in modular group algebras

Symmetric subgroups in modular group algebras

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