J. Group Theory 18 (2015), 887–904 DOI 10.1515/jgth-2015-0015 © de Gruyter 2015 The absolute center of finite groups Hangyang Meng and Xiuyun Guo Communicated by Evgenii I. Khukhro Abstract. In this paper we first investigate the relationship between the absolute center L.G/ and the Frattini subgroup ˆ.G/ for a finite group G, and then we describe the structure of finite groups G satisfying L.G/ ˆ.G/. For example, we prove that a finite group G is an F -group if and only if G=L.G/Ä is an F -group, where F is a saturated formation containing Z2. Next we determine all finite groups X such that X=L.X/ is isomorphic to Zp! Zp, where ! is a positive integer and Zp! is a cyclic group of order p! . 1 Introduction A group G is said to be capable if there exists a group H such that H=Z.H / is isomorphic to G. The study of capable groups plays a central role in various group- theoretical problems. Many mathematicians, such as Baer, Hall, Senior, Schur, Isaacs and so on, have investigated capable groups [1, 4, 8] and many interesting results have been given. For example, a classical result due to Schur states that if the central quotient G=Z.G/ of a group G is finite, then the commutator sub- group G0 is also finite, see [7]. In 1994, Hegarty [5] introduced the autocommutator subgroup K.G/ g 1g˛ g G; ˛ Aut.G/ D h W 2 2 i of a group G and its absolute center L.G/ g G g˛ g for all ˛ Aut.G/ : D ¹ 2 W D 2 º Furthermore, Hegarty proved an analogue of Schur’s theorem for the absolute center and the autocommutator subgroup of a group, that is, if G is a group such that G=L.G/ is finite, then K.G/ is also finite. Following Chaboksavar, Ghouchan and Saeedi [3], we say that a group G is autocapable if there exists a group M such that M=L.M / is isomorphic to G. In 1997, Hegarty [6] proved that for any finite autocapable group G there are finitely The research of the work was partially supported by the National Natural Science Foundation of China (11371237) and a grant of “The First-Class Discipline of Universities in Shanghai”. 888 H. Meng and X. Guo many finite groups M such that M=L.M / is isomorphic to G. However, it is not easy to see if a group G is autocapable. In 2014, Chaboksavar, Ghouchan and Saeedi proved that there is no group G such that G=L.G/ is isomorphic to Q8, and there are infinitely many finite and infinite non-autocapable groups. Now the following questions naturally arise. Can we describe the structure of a finite group G if G=L.G/ is known? Which finite groups X satisfy the equation X=L.X/ G for a given finite Š autocapable group G? In this paper we first investigate the relationship between L.G/ and the Frattini subgroup ˆ.G/ for a finite group G, and then we describe the structure of finite groups G satisfying L.G/ ˆ.G/. For example, we prove that a finite group G Ä is an F -group if and only if G=L.G/ is an F -group, where F is a saturated formation containing Z2 (Corollary 3.8). Next we determine the structure of the absolute center of all finite minimal non-abelian p-groups, and then we determine all finite groups X such that X=L.X/ is isomorphic to Zp! Zp, where ! is ! a positive integer and Zp! is a cyclic group of order p . 2 Preliminary results In this section we list some basic results, which are frequently used in this paper. Lemma 2.1. Let H1;:::;Hs be subgroups of a finite group G such that G H1 Hs: D Then L.G/ L.H1/ L.Hs/: Ä Moreover, if Hi is characteristic in G for i 1; : : : ; s, then D L.G/ L.H1/ L.Hs/: D Proof. Let ˛i be an automorphism of Hi for i 1; : : : ; s. Then we define a map D ˛ G G; x .h1; : : : ; hs/ ˛.x/ .h1; : : : ; hi 1; ˛i .hi /; hi 1; : : : ; hs/: W ! D 7! D C It is easy to see that ˛ is an automorphism of G. If x .h1; : : : ; hs/ L.G/, then D 2 it follows from .h1; : : : ; hs/ x ˛.x/ .h1; : : : ; ˛i .hi /; : : : ; hs/ D D D that ˛i .hi / hi and so hi L.Hi /. Thus L.G/ L.H1/ L.Hs/. D 2 Ä If Hi is characteristic in G for each i, then Aut.G/ Aut.H1/ Aut.Hs/, D which implies that L.H1/ L.Hs/ L.G/. Whence the result follows. Ä The absolute center of finite groups 889 Lemma 2.2 ([2, Theorem 3.2.]). Let H and K be subgroups of a finite group G such that G H K. If H and K have no common direct factor, then D Aut.G/ ® ˛ ˇ ˛ Aut.H /; ˇ hom.K; Z.H //; Š ı W 2 2 hom.H; Z.K//; ı Aut.K/¯: 2 2 Lemma 2.3. Let G be a finite group. Then L.G/ G if and only if G 1 or Z2. D Š Proof. This lemma is a special case of [3, Lemma 2.2]. 3 The absolute center and the Frattini subgroup Recall that a group H is said to be a direct factor of a group G if there exist subgroups K and T of G such that K H and G K T . In this section, we Š D investigate the relationship between L.G/ and the Frattini subgroup ˆ.G/ for a finite group G. The following lemma is the key lemma in this paper. Lemma 3.1. Let G be a finite group. If Z2 is not a direct factor of G, then L.G/ ˆ.G/: Ä Proof. Suppose that the lemma is false. Let G be a group such that Z2 is not a direct factor of G and L.G/ is not contained in ˆ.G/. Then there exists a maxi- mal subgroup M of G such that L.G/ M . Since L.G/ Z.G/, the maximality — Ä of M implies that G ML.G/ and M C G. Hence we assume that G=M p, D j j D where p is a prime number. Now we consider the following two cases. Case 1: p L.G/ M . Choose z L.G/ T M such that o.z/ p and fix j j \ j 2 D g L.G/ M . It is clear that 2 n p 1 G G Mgi : D i 0 D where “ ” means disjoint union of sets. For any x G, there exist unique m M t 2 2 and i 0; 1; : : : ; p 1 such that x mgi . Define a map 2 ¹ º D ˛ G G; x mgi mgi zi : W ! D 7! We claim that ˛ is an automorphism of G. In fact, for any two elements x mgi D and y ngj of G, where i; j 0; 1; : : : ; p 1 and m; n M , we see D 2 ¹ º 2 xy mgi ngj mngi j : D D C Since 0 i j < 2p 1 and gp M , we consider two possibilities. Ä C 2 890 H. Meng and X. Guo If i j p, then C ˛.xy/ ˛..mngp/gi j p/ D C .mngp/gi j pzi j p D C C .mgi zi /.ngj zj / D ˛.x/˛.y/: D If i j < p, then C ˛.xy/ ˛..mn/gi j / D C .mn/gi j zi j D C C .mgi zi /.ngj zj / D ˛.x/˛.y/: D In a word, ˛.xy/ ˛.x/˛.y/. In addition, for any element x. mgi / Ker.˛/, D D 2 in which m M and i 0; 1; : : : ; p 1 , 2 2 ¹ º 1 ˛.x/ ˛.mgi / mgi zi ; D D D which implies i 0 and m 1. Thus Ker.˛/ 1, and so the finiteness of G D D D implies that ˛ is an automorphism of G. Since g L.G/, it follows from the definition of L.G/ and ˛ that 2 g ˛.g/ ˛.1 g1/ gz: D D D Hence z 1, which is a contradiction. D Case 2: p ­ L.G/ T M . In this case, since j j \ p G=M ML.G/=M L.G/=L.G/ M ; D j j D j j D j j we see that p divides L.G/ and we may choose z L.G/ such that o.z/ p j j 2 D and z M . Hence G M; z M z : … D h i D h i The hypothesis of the lemma implies p > 2. It is clear that for any x G, there 2 exist unique m M and i 0; 1; : : : ; p 1 such that x mzi . Consider the 2 2 ¹ º D map ˇ G G; x. mzi / mz2i : W ! D 7! We claim that the map ˇ is an automorphism of G; the proof is similar to Case 1. Since z L.G/, it is clear that z ˛.z/ ˛.1 z1/ z2, a contradiction. 2 D D D The proof of the lemma is complete. The absolute center of finite groups 891 Remark 3.2. The hypothesis that Z2 is not a direct factor of G in Lemma 3.1 can not be removed. In fact, let a be an element of order 2, let H be a group of odd order and let G a H: D h i Then L.G/ L.