J. Group Theory 16 (2013), 275–298 DOI 10.1515/jgt-2012-0042 © de Gruyter 2013 Finite groups of fourth-power free order Cai Heng Li and Shouhong Qiao Communicated by Robert M. Guralnick Abstract. A characterization is given of finite groups of order indivisible by the fourth power of any prime, which shows that such a group has the following form: A B C D or .M .A B C //:E, where A; B; C and D are all nilpotent, M is perfect,W andW EW is abelian. Further,W W the semi-direct products involved in these groups are characterized. 1 Introduction A classical result, proved by Hölder [8] in 1893, says that a finite group of square- free order is a semi-direct product of a cyclic group by another cyclic group, namely, such a group is a semi-direct product of the form A B with A; B both cyclic. W Later on, some special small order groups were characterized; see for exam- ple [5, 11, 14, 15]. In particular, Dietrich and Eick [2] investigated the groups of cube-free order (that is, the order is not divisible by the cube of any prime). More recently, in [12], complementing the work of [2], it was shown that a finite non- nilpotent group of cube-free order has the form A B C or PSL.2; p/ .A B/; W W W where A; B and C are all abelian. A natural problem is to extend the cube-free case to the fourth-power-free case. A group is said to be of fourth-power-free order if its order is not divisible by the fourth power of any prime. The main purpose of this paper is to characterize such finite groups. Roughly speaking, it is shown that such a group has the form A B C D or .M .A B C //:E; W W W W W where A; B; C and D are all nilpotent, M is perfect, and E is abelian. Cai Heng Li is supported by NNSF of China (grants no. 11231008, 11171292) and an ARC Discovery Project grant. Shouhong Qiao acknowledges the support from the following grants: NNSF of China (11126098, 11201082), China Postdoctoral Science Foundation funded project (2012M521724) and FDYT (LYM11058) of Guangdong Province. The corresponding author is Shouhong Qiao. 276 C. H. Li and S. Qiao Before presenting the main result, we first state some special cases. As usual, by Gp with p prime we mean a Sylow p-subgroup of G. Corollary. Let G be a finite soluble group of fourth-power-free order. Then there exist non-trivial nilpotent groups L; M such that the following statements hold: (1) If . G ; 6/ 1, then G L, or L M . j j D D W (2) If . G ; 2/ 1, then G L, or L M , or L M G3. j j D D W W W (3) If . G ; 3/ 1, then G L, or L M , or L M G2. j j D D W W W Each case does occur. Here are examples with three nilpotent layers: 3 3 2 2 Z Z31 Z3 < AL.1; 5 / and Z D10 < AL.1; 11 /. 5 W W 11 W For the general case, we state the main results in two theorems. The first is for soluble groups. Theorem 1.1. Let G be a finite soluble group of fourth-power free order. Then one of the following holds, where L; M are non-identity nilpotent subgroups: (i) G L, or L M , D W (ii) G has three nilpotent layers, and has the form (i) L M G2 or L M G3 W W W W or 2 (ii) .L Z / .M Q8/ Z3, 3 W W (iii) G has four nilpotent layers, and has the form 2 L M G3 G2 or L .M Z / G3 Z2: W W W W 2 W W We remark that each case listed in the theorem indeed occurs. Here are a few of basic examples for three nilpotent layer groups: 2 2 3 1 2 3 2 Zp D2q; Zp S3; Zp 3 C Q8; Zp A4; Z3 Q8 Z3; W W W C W W W W 3 3 3 Z Q8 Z3; Z Z7 Z3; Z Z31 Z3: p W W 2 W W 5 W W The groups 3 3 3 3 2 .Z11 Z5/ S3 Z Z Z3 Z2 and Z S4 Z Z Z3 Z2 W o D 11 W 5 W W p W D p W 2 W W have four nilpotent layers. The semi-direct products appearing in the groups in Theorem 1.1 are essentially known, and we can easily see this in an indirect way: embedding such a group into Finite groups of fourth-power free order 277 a larger group, which is a direct product of some ‘basic’ groups, consisting of the following groups, where P divides p3 and Q divides q3, with p; q prime: j j j j P , P Q, W P Q R, with R dividing 23 or 33, W W j j 2 3 P Z R Z2, with R dividing 3 , W 2 W W j j P Q R S, with R dividing 33 and S dividing 23. W W W j j j j The structural information of these basic groups is described in Section 5. In par- ticular, the semi-direct products involved in these groups are determined. Corollary 1.2. Let G be a group of fourth-power free order. Then G B1 Bn; Ä where Bi is one of the basic groups described above. The next theorem characterizes the insoluble case. Theorem 1.3. Let G be an insoluble finite group of fourth-power-free order. Then G .M .A B C //:E, where A; B; C are nilpotent subgroups of odd order, D W W E is abelian, and M is a perfect subgroup satisfying one of the following W (i) M is almost quasi-simple: f A7, J1, PSL.2; 8/, or PSL.2; p /, where p is odd and f 3, 3 Ä 3:A6, 3:A7, SL.2; p/ with p 5, or SL.2; p / with p odd, (ii) M ASL.2; p/ Z2 SL.2; p/, where p 5, D D p W (iii) M F SL.2; 5/, where F is nilpotent with each Sylow r-subgroup isomor- D W2 2 3 phic to Z , Z Zr or Z , with either r 5, or r 1 .mod 10/, r r W r D Á ˙ (iv) M Z3 S, where m is square-free, and one of the following holds: D m W (a) S PSL.2; 7/, and r 11 and r3 1 .mod 7/ for each prime divi- D Á sor r of m, (b) S 3:A6, and r 1 or 19 .mod 30/ for each prime divisor r of m, D Á (c) S A5, and r 1 .mod 10/ for each prime divisor r of m. D Á ˙ The paper is organized as follows: In Section 2, we study automorphism groups of the groups of order pd with p prime and d 3. Then in Sections 2 and 3, we Ä treat the soluble case, and prove Theorem 1.1. Corollary 1.2 is proved in Section 4. In the final section, we prove Theorem 1.3. We conclude this section by giving some notation which will be frequently used. For a set of primes, 0 denotes the set of the primes not in . Let G be a group. We write G for a Hall -subgroup of G;Z.G/ denotes the centre of G, 278 C. H. Li and S. Qiao F.G/ denotes the Fitting subgroup of G, F .G/ the generalized Fitting subgroup of G (see, for example, [10, X, Section 13]), ˆ.G/ the Frattini subgroup of G. A semidirect product of a group A by a group B is denoted by A B, and then we W write A B C A .B C/ and A B C D A .B .C D//: W W D W W W W W D W W W 3 In what follows, Z denotes Z Z Z with l a positive integer, 3:A6 denotes l l l l a perfect group which has central factor group A6. 2 Automorphism groups of p-groups Let G be a group. The Fitting subgroup F F.G/ is the largest nilpotent normal WD subgroup of G. If G is soluble, then the Fitting subgroup satisfies CG.F / F . Ä Hence G=F G=.CG.F /F / Out.F / Out.F1/ Out.Ft /; D Ä D where F F1 Ft is such that each Fi is a Sylow pi -subgroup of F . D 3 Now Fi has order dividing pi , and thus structural information about subgroups of Out.Fi / is important in the study of G. Let P be a p-group of order at most p3. Then either P is abelian, or P is one of the two non-abelian groups of order p3. For p 2, the only non-abelian groups of 3 D order p 8 are Q8, the quaternion group, and D8, the dihedral group of order 8. D 3 2 If p is odd, then a non-abelian group of order p is Z Zp or Z 2 Zp. Note that p W p W 2 1 2 Z Zp has exponent p and is usually denoted by p , p W C 2 1 2 C 2 Zp Zp has exponent p and is denoted by p C .