A GENERALIZATION OF 2-BAER GROUPS Luise-Charlotte Kappe Department of Mathematical Sciences, Binghamton University 5 1 Binghamton, NY 13902-6000, USA 0 E-mail: [email protected] 2 b e Antonio Tortora F Dipartimento di Matematica, Universita` di Salerno 3 Via Giovanni Paolo II, 132 - 84084 - Fisciano (SA), Italy ] R E-mail: [email protected] G . h t Abstract a m A group in which all cyclic subgroups are 2-subnormal is called a 2- [ Baer group. The topic of this paper are generalized 2-Baer groups, 3 i.e. groups in which the non-2-subnormal cyclic subgroups generate a v propersubgroupofthegroup.Ifthissubgroupisnon-trivial,thegroup 0 iscalled ageneralized T -group.Inparticular,weprovidestructurere- 9 2 0 sults for such groups, investigate their nilpotency class and construct 3 examples of finite p-groups which are generalized T -groups. 0 2 . 1 2010 Mathematics Subject Classification: 20E15, 20F19, 20F45 0 5 Keywords: subnormal subgroup, 2-Baer group, Engel element 1 : v i 1 Introduction X r a Let nbea positive integer. Asubgroup H ofagroupG iscalledn-subnormal, denoted byH✁ G,if thereexist distinct subgroupsH = H,H ,...,H = G n 0 1 n such that H = H ✁H ✁...✁H = G. 0 1 n In a group of nilpotency class n, all subgroups are n-subnormal. Conversely, by a well-known result of Roseblade [23], a group with all subgroups n- subnormal is nilpotent of class bounded by a function of n. For n = 1, having all subgroups n-subnormal, hence normal, is equivalent to having all cyclic subgroups n-subnormal; but this is no longer the case if n ≥ 2 [18]. 1 A group G is called an n-Baer group if all of its cyclic subgroups are n-subnormal. It can be easily seen that every n-Baer group is (n+1)-Engel, i.e. [x, y] = 1 for all x,y ∈ G, where n+1 [x,1y] = [x,y] = x−1xy and [x,ky] = [[x,k−1y],y], k ≥ 2. We say x ∈ G is a right n-Engel element, if [x, g] = 1 for all g ∈ G, and a n left n-Engel element, if [g, x] = 1 for all g ∈ G. n Of course, the class of 1-Baer groups coincides with the familiar Dedekind groups. For the finite case these groups were classified by Dedekind in 1897 in [5], and for the case of infinite groups by Baer in [3]. Our interest in a generalization of n-Baer groups, in particular 2-Baer groups, is motivated by D.Cappitt’s research in a generalization of Dedekind groups [4]. He considers groups in which the non-normal subgroups of a group generate a proper sub- group of the group. Such groups were called generalized Dedekind groups. In [20] it was shown that the subgroup generated by all non-normal cyclic sub- groups coincides with the subgroup generated by all non-normal subgroups. Generalized Dedekind groups which are not Dedekind groups are called gen- eralized Hamiltonian groups. Recalling that a Dedekind group is an n-Baer group for n = 1, we want to extend Cappitt’s concept of generalized Dedekind groups for n > 1. For any group G, let T (G) = hx ∈ G|hxi ⋪ Gi. n n If all cyclic subgroups are n-subnormal in G, i.e. G is an n-Baer group, we define T (G) = 1. Notice that T (G) is a characteristic subgroup of G and n n G/T (G) is an n-Baer group. n In light of [4] and [14], we generalize the concept of an n-Baer group as follows. Definition 1.1. If G is a group with T (G) 6= G, then G is called a general- n ized n-Baer group, and, if in addition T (G) is non-trivial, then G is called n a generalized T -group. n The class of generalized n-Baer groups and the class of generalized T -groups n will be denoted by GB and GT , respectively. n n In [4] it was shown that a group in GB is either abelian or torsion and 1 nilpotent of class 2. The question arises, if there are any restrictions on the nilpotency class of generalized n-Baer groups. It was shown in [11] and [19] that 2-Baer groups are of nilpotency class at most 3. However, as it was shown in[10], n-Baer groupswithn ≥ 3 arenot necessarily nilpotent. Thus it appears to be natural to restrict our attention to generalized 2-Baer groups. As we will see, our results are, for the most part, analogue to those for 2 generalized Dedekind groups, i.e. generalized 1-Baer groups. Here are some of the details of our investigations. Foranon-torsiongroup,which isageneralized 2-Baergroup,wewill show in Theorem 2.5 that such a group is actually 2-Baer or, equivalently, 2-Engel. This is the direct analogue of Cappitt’s result in [4]. For torsion groups which are generalized 1-Baer groups, Cappitt [4] shows that such groups are either 1-Baer groups or a direct product of a p-group in GB and another factor 1 which is 1-Baer without p-torsion. Our result, presented in Theorem 2.6, is exactly the same, replacing 1-Baer by 2-Baer, and GB by GB . 1 2 However when it comes to restrictions on the nilpotency class of torsion groups in GT , our result differs from the one obtained by Cappitt in [4], 2 where it was shown that torsion groups in GT are of nilpotency class 2. 1 For metabelian p-groups in GT with p > 2 we can show that such groups 2 are nilpotent of class exactly 3 (Theorem 3.1). In Example 4.1 we present a metabelian 2-group obtained by GAP [9], which is in GT but of nilpotency 2 class 4, showing that Theorem 3.1 cannot be extended to 2-groups. In Ex- ample 4.3, for any prime p, we provide metabelian p-groups which are in GT 2 and have nilpotency class 3. Using results by Endimioni in [6, 7, 8], it can be shown that locally finite p-groups in GT are soluble if p 6= 5, and for p > 5 those groups are nilpotent 2 of bounded class (see Theorem 2.8). The exact bound of the nilpotency class for groups in GT remains an open question. 2 2 Structure Results In this section we provide some structure results for generalized 2-Baer groups. Asalreadypointedoutintheintroduction,theresultsfornon-torsion and torsion groups alike are analogue to those determined by Cappitt [4] for the case of generalized 1-Baer groups. We start with some preparatory lem- mas. Lemma 2.1. Let G ∈ GB . 2 (i) If x ∈ G\T (G) and H is a subgroup of G containing x, then H ∈ GB . 2 2 (ii) If N is a normal subgroup of G contained in T (G), then G/N ∈ GB . 2 2 Proof. SinceT (H) ≤ T (G)andT (G/N) ≤ T (G)/N,wehavex ∈ H\T (H) 2 2 2 2 2 and T (G/N) 6= G/N, respectively. 2 Given an element x of a group G, throughout the paper, we denote by hxGi the normal closure of x in G, i.e. the smallest normal subgroup of G containing x. 3 Lemma 2.2. Let G ∈ GB and x ∈ G\T (G). Then hxGi is nilpotent of class 2 2 at most 2. In particular, x is a left 3-Engel element. Proof. Clearly, hxi✁hxGi and N (hxi)/C (x) is abelian. Denote by H the G G commutator subgroup of hxGi. Then hxGiC (x)/C (x) is abelian and there- G G foreH ≤ C (x). Let a ∈ H andg ∈ G. Nowag−1 ∈ H,so that [ag−1,x] = 1 or G [a,xg] = 1. This means that H ≤ C (xg), for all g ∈ G. Hence, H ≤ Z(hxGi) G and hxGi is nilpotent of class ≤ 2. Proposition 2.3. Let G ∈ GB . Then, for any d ≥ 1, every d-generator 2 subgroup of G is nilpotent of class at most 2(d+1). Proof. Let H = hh ,...,h i be a subgroup of G and x ∈ G\T (G). Con- 1 d 2 sider K = hh ,...,h ,xi. Next we want to show that we can find elements 1 d k ,...,k such that K = hk ,...,k ,xi and k ∈ G\T (G). For any 1 ≤ i ≤ d 1 d 1 d i 2 we set k = h , ifh ∈ G\T (G)andk = h xotherwise. Soeachk ∈ G\T (G) i i i 2 i i i 2 and thus K is contained in the subgroup hkGi...hkGihxGi, which is nilpotent 1 d of class ≤ 2(d+1) by Lemma 2.2. Lemma 2.4. Let G ∈ GB and x an element of G\T (G) of infinite order. 2 2 Then: (i) For any g ∈ G, there exists an integer n = n(g) ≥ 0 such that xng is an element of G\T (G) of infinite order; 2 (ii) hxGi is abelian and x is a right 2-Engel element. Proof. (i) Given an arbitrary element g of G, we consider first the case that g ∈ T (G). Thus xg ∈ G\T (G). Moreover, by Lemma 2.2, hx,gi ≤ 2 2 hxGih(xg)Gi and hence hx,gi is nilpotent. If xg has finite order, this im- plies that x2g and x3g have infinite order. However, if x2g ∈ T (G), then 2 x3g ∈ G\T (G). Next let g ∈ G\T (G). Obviously, we may assume that g 2 2 has finite order. If xg ∈ G\T (G), our claim follows as before. Thus we may 2 assume xg ∈ G\T (G). By Lemma 2.2 it follows that hx,gi is nilpotent and 2 so xg has infinite order. (ii) We prove that x ∈ Z(hxGi). Suppose to the contrary and let a ∈ hxGi such that xa 6= x. Since hxi ✁ hxGi, we have xa = xm for some integer m 6= 0,1. Similarly, xa−1 = xm′ with m′ 6= 0,1. Then xmm′ = x, or xmm′−1 = 1. This gives m = m′ = −1 and so xa = x−1. We conclude [a,x] = x2. Lemma 2.2 implies that x2 ∈ Z(hxGi) and therefore 1 = [a,x2] = [a,x]2 = x4, a contradiction. Hence, x ∈ Z(hxGi) and hxGi is abelian. In particular, [g,x,x] = 1 for all g ∈ G. On the other hand, by (i), for any g ∈ G there 4 exists k ≥ 0 such that xkg is an element of G\T (G) of infinite order. Thus 2 1 = [x,xkg,xkg] = [x,g,xkg] = [x,g,g], which shows that x is a right 2-Engel element. In [11], Heineken already showed that in the class of non-torsion groups, 2-Baer and 2-Engel are equivalent properties. We extend this result here for generalized 2-Baer groups. Theorem 2.5. Let G be a non-torsion group. Then the following are equiv- alent: (i) G is a 2-Baer group; (ii) G is a generalized 2-Baer group; (iii) G is a 2-Engel group. Proof. Denote by R (G) = {x ∈ G|[x,g,g] = 1 for all g ∈ G} 2 the set of all right 2-Engel elements of G. Since the normal closure of any 2-Engel element is abelian [17] (see also [21, Theorem 7.13]), and hence every cyclic subgroup is 2-subnormal, it is enough to show that (ii) implies (iii). Let a,b ∈ G and take x ∈ G\T (G) of infinite order. Lemma 2.4 implies 2 that x ∈ R (G) and there exist m,n ≥ 0 such that xma,xnb ∈ R (G). 2 2 But R (G) is a subgroup of G [15] (see also [21, Corollary 1, p. 44]), so 2 that xm is also an element of R (G). On the other hand, by Lemma 2.4, 2 ha,b,xi = hxma,xnb,xi is nilpotent of class ≤ 3. Thus we get 1 = [xma,b,b] = [xm,b,b][xm,b,a,b][a,b,b] = [a,b,b]. This proves that G is 2-Engel. Turning now to the class of torsion groups which are generalized 2-Baer groups,ourcharacterizationisparalleltotheoneofgeneralized1-Baergroups [4]. Theorem 2.6. Let G be a torsion group and assume that G is a generalized 2-Baer group. Then G = H ×K where H is a locally finite p-group in GB 2 for some prime p and K is a 2-Baer group without elements of order p. 5 Proof. By Proposition 2.3, the group G is locally nilpotent, hence G is the direct product of its Sylow p-subgroups. Let H be a Sylow p-subgroup of G p and suppose T2(Hp) = Hp. Thus G = Hp×Kp′, where Kp′ is the complement of Hp in G. Let h ∈ Hp and k ∈ Kp′, with hhi ⋪2 Hp. Clearly, if hhki✁2 G, then hhi = hhki∩H ✁ H . Therefore, we have necessarily hk ∈ T (G). It p 2 p 2 follows that HpKp′ = T2(Hp)Kp′ ≤ T2(G) and G = T2(G), a contradiction. ThismeansthatHp ∈ GB2 forallprimesp.Ontheotherhand,ifT2(Kp′) 6= 1, we get asbeforeHp ≤ HpT2(Kp′) ≤ T2(G)andG = T2(G). Hence thereexists a prime p such that G = H × K where H is a p-group in GB and K is a 2 2-Baer group without elements of order p. Corollary 2.7. A group G is a generalized T -group if and only if G = 2 H ×K, where H is a locally finite p-group in GT for some prime p and K 2 is a 2-Baer torsion group without elements of order p. Proof. Assume G = H ×K, where H is a p-group and K is a torsion group without elements of order p. Let x = hk, with h ∈ H and k ∈ K, such that ′ ′ hxi ⋪ hxGi. Then hhi ⋪ hhHi, because otherwise x(xhk ) ∈ hh,ki = hxi for ′ ′ all h ∈ H and k ∈ K. So h ∈ T (H) and x ∈ T (H)K. This proves that 2 2 T (G) ≤ T (H) × K. Similarly we have T (G) ≤ H × T (K). Thus, if G is 2 2 2 2 a generalized T -group, the claim follows from Theorems 2.5 and 2.6. The 2 converse is trivial. Let c and d be positive integers. A group G is said to be a group of type (d,c) if every d-generator subgroup of G is nilpotent of class at most c. In this context we are interested in groups of type (d,3d−3) and of type (d,cd), that were investigated by Endimioni in [7, 8] and [6], respectively. Theorem 2.8. Let G be a locally finite p-group and assume that G is a generalized 2-Baer group. (i) If p 6= 5, then G is soluble. (ii) If p > 5, then G is nilpotent of bounded class. (iii) If p = 5 and G is soluble, then G is nilpotent of class bounded by a function depending on the derived length. Proof. By Proposition 2.3, the group G is of type (d,3d − 3) with d = 5, and of type (d,3d) with d = 2. In particular G is 6-Engel. If p 6= 2,5 then G is soluble of bounded derived length, by [7, Proposition 3(ii)]. On the other hand, if p = 2, then G is again soluble, as shown in [2, p. 2830] (see also [8]). However, when p > 5, the main theorem of [7] guarantees that G is nilpotent of bounded nilpotency class. Finally, if p = 5 and G is soluble, we get (iii) by [6, Corollary 2]. 6 3 Metabelian generalized T -groups 2 As Theorem 2.8of the last section suggests, there might bea universal bound for the nilpotency class of p-groups in GT provided p > 5. But so far we have 2 not been able to find an explicit bound for their nilpotency class. However in the case of metabelian groups we have been able to show that p-groups in GT have nilpotency class 3, provided that p > 2. In the last section we will 2 provide an example of a 2-group in GT of nilpotency class 4, showing that 2 our restriction to odd primes is necessary. ′ Let G be a metabelian group and denote by G its commutator subgroup. ′ For integers m,n with n > 0 and for all x,y,z ∈ G,c ∈ G, in the sequel we will use freely the identities [c,x,y] = [c,y,x], [xy, z] = [x, z][x, z,y][y, z] n n n n and, if G is nilpotent of class 3, [xm,y,z] = [x,ym,z] = [x,y,zm] = [x,y,z]m. Now we are ready to state and prove the main result of this section. Theorem 3.1. Let G be a metabelian p-group with p an odd prime. If G ∈ GT , then G is nilpotent of class 3. 2 Proof. By Theorem 7.36 (i) in [21], it is enough to show that G is a 3-Engel group. For this it suffices to show that for every pair of elements x,y in G we have [x, y] = 1 and [y, x] = 1. We consider three cases: x and y are both in 3 3 G\T (G), only one of the elements is in G\T (G), and x and y are both in 2 2 T (G). Set H = hx,yi. 2 In the first case, we observe by Lemma 2.2 that [y, x] = 1 and [x, y] = 1. 3 3 Nowconsider thesecondcase, wherewe canassume without lossofgenerality that x ∈ G\T (G) and y ∈ T (G). We observe that also xy ∈ G\T (G) 2 2 2 and hence hxGih(xy)Gi is a group of nilpotency class 4 at most and H ≤ hxGih(xy)Gi. By Lemma 2.2 we have [y, x] = 1 and [x, xy] = 1. To prove 3 3 our claim, we need to show [x, y] = 1. Since H is of class 4 at most, we 3 obtain by linear expansion that 1 = [x, xy] = [x, y][x,y,y,x][x,y,x,y]. (3.1.1) 3 3 Substituting x−1 for x into (3.1.1) and again expanding linearly yields 1 = [x−1, x−1y] = [x, y]−1[x,y,y,x][x,y,x,y]. (3.1.2) 3 3 7 Multiplying (3.1.1) by the inverse of (3.1.2) leads to [x, y]2 = 1. Since G has 3 no involutions, we conclude [x, y] = 1, the desired result. 3 Before we prove the third case, i.e. x and y are both in T (G), we want 2 to show that H is nilpotent of class 3 at most, provided that not both x and y are in T (G). Now (3.1.1) together with [x, y] = 1 yields that 2 3 1 = [x,y,y,x][x,y,x,y]. By III, 2.12 (b) in [13], we have that any 2-generator group of class 4 is metabelian and thus [x,y,y,x] = [x,y,x,y]. This together with the fact that our group has no involutions yields 1 = [x,y,y,x]. Now we observe that γ (H) = h[y, x],[x, y],[x,y,y,x]i. 4 3 3 But we showed that each of the generators is the identity in H. We conclude γ (H) = 1 and H is of class 3. 4 To prove the remaining case, i.e. x,y ∈ T (G) implies [x, y] = [y, x] = 1, 2 3 3 we recallthatGismetabelian. Letz ∈ G\T (G)andx,y ∈ T (G).Obviously, 2 2 xz ∈ G\T (G). Then the subgroups hy,zi and hy,xzi are nilpotent of class 2 3, by the previous argument. It follows that 1 = [xz, y] = [x, y][x, y,z][z, y] = [x, y][x, y,z]. 3 3 3 3 3 3 By commuting with z, we get 1 = [x, y,z][x, y,z,z]. Commuting again 3 3 with z yields 1 = [x, y,z,z]. Thus 1 = [x, y,z] and consequently 1 = [x, y]. 3 3 3 Similarly we obtain [y, x] = 1. We conclude that G is 3-Engel. 3 4 Examples and Counterexamples In this section we provide various examples and counterexamples in sup- port of claims made earlier in this paper. In Theorem 3.1 it was shown that metabelian p-groups in GT , p an odd prime, have nilpotency class exactly 2 3. Our first example shows that this result cannot be extended to 2-groups. Our claims have been verified by GAP [9]. Example 4.1. Let G = hx,y|x16 = y16 = 1,(xy−1)2 = [x,y]4 = 1,[x,y,x] = x4,[x,y,y] = y4i. Then G is a metabelian group of order 27 and nilpotency class 4. Moreover |T (G)| = 26, thus G is a generalized T -group. 2 2 Forournextexampleweneedthefollowingexpansionformulaformetabelian groups which can be found in [12]. 8 Lemma 4.2. Let G be a metabelian group, x ∈ G and n an integer. Then (xy−1)n = xn( Y [x,iy,jx](i+nj+1))y−n. (4.2.1) 0<i+j<n The next example provides a p-group G with 1 6= T (G) 6= G for any 2 prime p. Example 4.3. For any prime p, let G = hx,y|xp3 = yp3 = [x,y]p = [x,y,x] = 1,[x,y,y] = xp2 = yp2i. Then G is a generalized T -group of nilpotency class 3 and of order p6 with 2 |T (G)| = p5. 2 Proof. We divide the proof into three steps. In the first one we will show that the group G is nilpotent of class 3 and of order p6. Let N = h[x,y],xp2i. We have N ✁G and G/N abelian, so that G′ ≤ N. ′ It follows that G = N ≤ Z (G) and G is nilpotent of class 3 (see also 2 [1, Proposition 3.1]). In particular, G is finite and, by the Burnside Basis Theorem [22, 5.3.2], Frat(G) = G′Gp = hxp,yp,[x,y]i. Thus every g ∈ G can be written as g = xmynz, where 0 ≤ m,n ≤ p−1 and z ∈ Frat(G). Since |Frat(G)| = p4, we deduce that |G| = p6. In our next step we will provide a necessary and sufficient condition for g ∈ G, such that hgi is 2-subnormal inG, as follows: Let g = xmynz ∈ G with 0 ≤ m,n ≤ p−1 and z ∈ Frat(G). Then hgi✁ G if and only if m ≡ n ≡ 0 2 (modp) or m+n 6≡ 0 (modp). This can be seen as follows. Clearly, we have Frat(G) ≤ Z (G) and 2 ′ exp(G) = p. Then [x,g,g] = [x,yn,yn] = xn2p2 and [y,g,g] = [y,xm,yn] = x−mnp2. Moreover, applying (4.2.1), we get gp2 = xmp2ynp2 = x(m+n)p2. Hence [x,g,g] ∈ hgi and [y,g,g]∈ hgi if and only if xn2p2 ∈ hx(m+n)p2i and x−mnp2 ∈ hx(m+n)p2i. This is equivalent to the existence of integers α and β satisfying the following congruences: n2 ≡ α(m+n) (modp); (4.3.1) 9 −mn ≡ β(m+n) (modp). (4.3.2) Now let hgi✁ G. This gives [x,g,g] ∈ hgi and [y,g,g] ∈ hgi. Therefore 2 (4.3.1) and (4.3.2) are satisfied for some α and β. Assuming m + n ≡ 0 (modp), it follows that m ≡ n ≡ 0 (modp). Hence, either m ≡ n ≡ 0 (modp) or m+n 6≡ 0 (modp). Conversely, let h = xm′yn′z′ be an arbitrary element of G where 0 ≤ ′ ′ ′ m,n ≤ p−1 and z ∈ Frat(G). It is easy to see that [h,g,g] = [xm′yn′,g,g] = [xm′,g,g][yn′,g,g] = [x,g,g]m′[y,g,g]n′. The claim will follow once we have shown that [h,g,g] ∈ hgi. If m ≡ n ≡ 0 (modp), then g ∈ Frat(G) and [h,g,g] = 1. Suppose m+n 6≡ 0 (modp). In this case there always exist integers α and β satisfying (4.3.1) and (4.3.2). Thus, by the above, [x,g,g],[y,g,g]∈ hgi and so is [h,g,g]. Next we will show that the group G is a generalized T -group by estab- 2 lishing that |T (G)| = p5. Let g = xmynz ∈ G. Then, by the above, we have 2 hgi ⋪ G if and only if m+n ≡ 0 (modp) and mn 6≡ 0 (modp). It follows 2 T (G) = hg|g = xmyp−mz,0 < m ≤ p−1,z ∈ Frat(G)i. 2 On the other hand, by (4.2.1), we have xmyp−mz = (xyp−1)mz′ for a suitable ′ z ∈ Frat(G). It follows that T (G) = hxyp−1,xp,yp,[x,y]i. 2 Notice also that xyp−1 ∈/ Frat(G), otherwise 1 = [xyp−1,y,y] and conse- quently 1 = [x,y,y]. However, again by (4.2.1), it follows (xyp−1)p = xp[x,y1−p](p2)[x,y1−p,y1−p](p3)y(p−1)p ∈ Frat(G) and (xyp−1)p2 = xp2y(p−1)p2 = xp2x(p−1)p2 = xp3 = 1. Since |Frat(G)| = p4, we conclude that |T (G)| = p5. This proves that G ∈ 2 GT . 2 References [1] A. Abdollahi, R. Brandl and A. Tortora, Groups generated by a finite Engel set, J. Algebra 347 (2011), 53–59. [2] A. Abdollahi and G. Traustason, On locally finite p-groups satisfying an Engel condition, Proc. Amer. Math. Soc. 130 (2002), no. 10, 2827–2836. 10