View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Florence Research Algebra Colloquium 11:2 (2004) 181{190 Algebra Colloquium °c AMSS CAS 2004 p-Groups with All the Elements of Order p in the Center¤ Daniela Bubboloni Gabriella Corsi Tani DIMAD Universita' di Firenze via C. Lombroso 6/17, I-50134 Firenze, Italy E-mail: [email protected]¯.it [email protected]¯.it Received 26 April 2001 Communicated by C.K. Gupta Abstract. Resting on a suitable base of the quotients of the ¸i-series for the free groups on r generators, we get, for p odd, a class of TH-p-groups (the groups n in the title) Gr with arbitrary large derived length. We prove that every TH-p- n n group G with r generators and exponent p is a quotient of Gr and a product m of m cyclic groups, where p = j­1(G)j: At last we describe the TH-p-groups of exponent p2: 2000 Mathematics Subject Classi¯cation: 20D15 Keywords: p-groups, central series, free groups 1 Introduction Let G be a p-group and p an odd prime. We denote by ­i(G) the subgroup of G generated by the elements of order dividing pi, and we call G a TH- p-group if all its elements of order p are central, i.e., ­1(G) · Z(G): This name was introduced by the authors in [2] in acknowledgment to Thompson who ¯rst obtained some classical results for the number of generators of these groups (see [4, III, 12.2]). In [2], it was shown that several properties of the regular p-groups hold also for the class of TH-p-groups. There we 2 characterized the TH-p-groups G with j­1(G)j = p and exhibited some other examples of TH-p-groups obtaining only metabelian groups. In this paper, following a suggestion of C.M. Scoppola, we construct a n class of TH-p-groups Gr (see De¯nition 3.1) with arbitrary large derived length (see Theorems 3.2 and 3.3). This construction rests on the properties of the central series ¸i(Fr) of the free group Fr on r generators and on the behaviour of a particular base ¤Supported by M.U.R.S.T. and C.N.R. 182 D. Bubboloni, G. Corsi Tani of the elementary abelian quotients ¸i(Fr)=¸i+1(Fr) (see Theorem 2.5(c)). The explicit determination of a base for ¸i(Fr)=¸i+1(Fr) has interest in itself and we devote Section 2 to this, resting on methods and results in [5] and [1]. Similar goals for the central series ·n(Fr); the Jennings{Lazard{Zassen- haus series of G, were reached by C.M. Scoppola in [6] (compare Lemma 1.11 and Proposition 2.5 in [6] with (a) and (c) of Theorem 2.5). In Section 4, we observe that each TH-p-group with r generators and n n exponent p is a quotient of the group Gr (Theorem 4.1). Moreover, we obtain some new general result about the structure of TH-p-groups which turn out to be a suitable product of cyclic groups (Theorem 4.3). Finally, in Section 5, we describe the TH-p-groups of exponent p2: The notation is standard. We indicate by γi(G) the i-th term of the lower central series of a group G: Throughout this paper, p will be always an odd prime. 2 The ¸i-Series of the Free Groups We recall the construction of the central series ¸i(G) of a group G and the properties of this series which we intend to use, collecting them from [5] and [1]. De¯nition 2.1. For any group G, put pi¡1 pi¡2 ¸i(G) := γ1(G) γ2(G) ¢ ¢ ¢ γi(G)(i ¸ 1): Thus, ¸i(G) is a characteristic subgroup of G and G = ¸1(G) ¸ ¸2(G) ¸ ¢ ¢ ¢ ¸ ¸n(G) ¸ ¢ ¢ ¢ : Theorem 2.2. [5, 1] For any i 2 N; the following properties hold: (a) [¸i(G); ¸j(G)] · ¸i+j(G); p (b) ¸i(G) = [¸i¡1(G);G] ¸i¡1(G) ; pi¡1 (c) [¸i(G);G] = γ2(G) ¢ ¢ ¢ γi+1(G); pi¡j (d) if G/γj(G) is torsion free, then ¸i(G) \ γj(G) = γj(G) ¢ ¢ ¢ γi(G); pj (e) ¸i(G) · ¸i+j(G); (f) the ¸i-series is central and ¸i(G)=¸i+1(G) is an elementary abelian p-group. De¯nition 2.3. Let Fr be the free group on r free generators x1; x2; : : : ; xr, p and A = (a1; a2; : : : ; an) be an ordered subset of Fr. We denote by A the p p p p ordered subset A = (a1; a2; : : : ; an). Moreover, if H £ Fr, we denote with A mod H the ordered subset (a1H; a2H; : : : ; anH) of Fr=H. For brevity, we will often write ¸i and γi instead of ¸i(Fr) and γi(Fr), respectively. p-Groups with Elements of Order p in Center 183 Lemma 2.4. Let a1; a2; : : : ; an 2 ¸i¡1 (i ¸ 2): Then p p p p (a1a2 ¢ ¢ ¢ an) ´ (a1 a2 ¢ ¢ ¢ an ) mod ¸i+1: Proof. Let a1; a2 2 ¸i¡1: By the Hall{Petrescu formula, there exist elements ck 2 γk(¸i¡1)(k = 2; : : : ; p) such that (p) p p p p 2 (p¡1) a1 a2 = (a1a2) c2 ¢ ¢ ¢ cp¡1 cp: Since i; k ¸ 2, we get ¸k(i¡1) · ¸i, and by (a) and (e) of Theorem 2.2, we p obtain γk(¸i) · ¸ki and ¸i · ¸i+1; so whenever k · p ¡ 1; we get (p) p p p ck k 2 γk(¸i¡1) · ¸k(i¡1) · ¸i · ¸i+1: For k = p, since i ¸ 2 and p ¸ 3 imply p(i ¡ 1) ¸ i + 1, we get cp 2 γp(¸i¡1) · ¸p(i¡1) · ¸i+1: Hence, p p p a1 a2 ´ (a1a2) mod ¸i+1: Now by induction on n; the lemma follows at once. 2 Theorem 2.5. (a) ¸i¡1=¸i may be embedded into ¸i=¸i+1: Moreover, there exists a base p Ai¡1 mod ¸i of ¸i¡1=¸i such that Ai¡1 mod ¸i+1 is independent in ¸i=¸i+1. (b) For each i, let Ai mod γi+1 be a base of γi/γi+1: Then pi¡1 pi¡2 (A1 [A2 [ ¢ ¢ ¢ [ Ai) mod ¸i+1 is a base for ¸i=¸i+1: (c) Let Ci denote the set of the basic commutators of weight i in a ¯xed sequence, then pi¡1 pi¡2 Bi := (C1 [C2 [ ¢ ¢ ¢ [ Ci) mod ¸i+1 is a base of ¸i=¸i+1: p (d) The map 'i : ¸i¡1 ! ¸i=¸i+1 de¯ned by x'i = x ¸i+1 is a homomor- phism and ker 'i = ¸i. Proof. (a) By a well-known result of Blackburn [5, VIII, 1.9b)], since p is odd, there is an isomorphism γ1 γi ®i : p £ ¢ ¢ ¢ £ p ¡! ¸i=¸i+1 γ1 γ2 γi γi+1 184 D. Bubboloni, G. Corsi Tani pi¡1 pi¡2 given by (¹a1; a¹2;:::; a¹i)®i = a1 a2 ¢ ¢ ¢ ai¸i+1: If ¹ is the natural im- γ1 γi¡1 γ1 γi mersion of p £ ¢ ¢ ¢ £ p into p £ ¢ ¢ ¢ £ p , then the map γ1 γ2 γi¡1 γi γ1 γ2 γi γi+1 ¡1 ¼i = ®i¡1 ¹®i : ¸i¡1=¸i ¡! ¸i=¸i+1 is a monomorphism. γj Leta ¹j 2 p and x = (1;:::; 1; a¹j; 1;:::; 1)®i¡1 2 ¸i¡1=¸i: Then γj γj+1 pi¡j¡1 x = aj ¸i and we have pi¡j x¼i = aj ¸i+1 2 ¸i=¸i+1: p p Now if Cj mod γj γj+1 is a base of γj/γj γj+1; then i[¡1 i[¡1 p pi¡1¡j ((Cj mod γj γj+1)®i¡1) = (Cj mod ¸i) j=1 j=1 is a base, say Ai¡1 mod ¸i of ¸i¡1=¸i: Hence, we obtain that i[¡1 pi¡j p (Ai¡1 mod ¸i)¼i = (Cj mod ¸i+1) = Ai¡1 mod ¸i+1 j=1 is a base of Im ¼i. (b) First of all, we observe that if Ai mod γi+1 is a base of γi/γi+1, p since γi/γi+1 is a torsion-free abelian group, Ai mod γi γi+1 is a base of p the elementary abelian group γi/γi γi+1. p p Next the base (A1 mod γ1 γ2) [ ¢ ¢ ¢ [ (Ai mod γi γi+1) of γ1 γi p £ ¢ ¢ ¢ £ p γ1 γ2 γi γi+1 i¡1 p pi¡2 is taken onto the base (A1 [A2 [ ¢ ¢ ¢ [ Ai) mod ¸i+1 of ¸i=¸i+1 by the isomorphism ®i which we mentioned in the proof of (a). (c) By the well-known Hall's basis theorem, Ci mod γi+1 is a base of γi/γi+1 (see [3, Chapter 11]). Thus, (c) follows from (b). ¸i p (d) By Lemma 2.4, the map 'i : ¸i¡1 ! given by x'i = x ¸i+1 is ¸i+1 a homomorphism. We prove ker 'i = ¸i. Clearly, ¸i · ker 'i, and we only need to show p that for x 2 ¸i¡1, the condition x 2 ¸i+1 implies x 2 ¸i. By (a), we can choose a base Ai¡1 mod ¸i for ¸i¡1=¸i with Ai¡1 = (a1; a2; : : : ; at) p ®1 ®2 ®t such that Ai¡1 mod ¸i+1 is independent in ¸i=¸i+1.