JOURNAL OF ALGEBRA 181, 223᎐234Ž. 1996 ARTICLE NO. 0117
Some Problems on Dedekind-Type Groups
Ramji Lal
Department of Mathematics, Uni¨ersity of Allahabad, Allahabad 211002, India
Communicated by George Glauberman
Received March 21, 1994 View metadata, citation and similar papers at core.ac.uk brought to you by CORE
provided by Elsevier - Publisher Connector 1. INTRODUCTION
Knowing the properties of subgroups of a group, what can we say about the group? This problem has been of consistent interest in the theory of groups from the beginning of the century. Miller and Morenowx 16 studied groups all of whose proper subgroups are abelian. Schmidtwx 29 and Golfondwx 10 determined all finite groups with all proper subgroups nilpo- tent. The above authors were interested in groups all of whose proper subgroups have a given property, no matter how the subgroups are embedded in the group. The characterisation of groups all of whose subgroups are embedded in a given manner also has a long history. The results of Dedekindwx 6 and Baer wx 2 characterize Dedekind groupsŽ groups all of whose subgroups are normal. . Rosebladewx 28 studied groups all of whose subgroups are subnormal. The study of the structure of groups which have subgroups of mixed type Ži.e., a subgroup of the group either has a prescribed property or is embedded in a given manner. has drawn the interest of several authors. Nagrebeckiiwx 17᎐19 , Romalis and Sesekinwx 25᎐27 , Cernikovˇ wx 5 , Garascukˇ wx9 , and Mahanevw 15 x were interested in meta-Hamiltonian groupsŽ groups all of whose subgroups are normal or abelian. . Phillips and Wilsonwx 21 have determined groupsŽ. with some finiteness condition in which every subgroup is either serial or abelian. Bruno and Phillipswx 4 deal with non-locally nilpotent groupsŽ. with some finiteness condition in which every subgroup is either normal or locally nilpotent. The transversals of a subgroup of a group determine the embedding of the subgroup in the group. The object of this paper is to introduce new types of subgroups by specifying the properties of transversals and initiate
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0021-8693r96 $18.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 224 RAMJI LAL a programme of research dealing with the problems of the above type in relation to these types of subgroups. Let H be a subgroup of a group G and S a right transversal of H in G which contains the identity. Then S is a right quasigroup with identity with respect to the binary operation ( defined by Ä4x( y s S l Hxy; x, y g S. Conversely, we have
THEOREM wx22 . Let S be a right quasigroup with identity. Then there is a S group G containing S as a right trans¨ersal such that gi¨en any other group G containing S as a right trans¨ersal, there exists a unique homomorphism from GtoGS which is the identity map on S. If a subgroup H is normal in G, then all the transversals of H in G are isomorphic as right quasigroupsŽ they are isomorphic to the quotient group GrH.. Let us call a subgroup H a perfectly stable subgroup if all the right transversals of H in G are isomorphic as right quasigroups. For finite groups we have
THEOREM wx23 . E¨ery perfectly stable subgroup of a finite group is normal. For infinite groups we do not know the answer to the preceding result. Consider the subgroup ²:S of a group G generated by a right transver- ²: sal S of a subgroup H of a group G. Let HSSs S l H. Then H s 1 ²ÄxyŽ. x( yxy<,ygS4:Ž, where ( is the induced binary operation on S as .²: r discussed earlier and HSS s S. Identifying S with the set Gr H of right cosets of H in G, we obtain the permutation representation : Ž.ŽÄ Ž.Ž.4 .Ž. GªSym S gxsSlHxg; g g G, x g S . Let GSSs H . Then GS is the group torsionwx 22Ž. or associator of the induced right quasigroup structure on S Ždepends only on the right quasigroup structure on S.. Clearly, a subgroup H of a group G is normal if and only if GS is trivial for every right transversal S of H in G.
DEFINITION 1.1. A subgroup H of a group G will be called a stable subgroup of G if for any right transversals S and S of H in G, G is 12 S1 isomorphic to G . S2 Thus all normal subgroups are stable. Every subgroup of order two of a non-abelian simple group is stable. In fact if G is a non-abelian simple group of order n and H is a subgroup of order k such that k!- 2n, then H is a stable subgroup of G. In particular, every finite group can be embedded as a stable subgroup into a finite simple group. Tarski groups Žtwo generator infinite simple groups all of whose proper subgroups are cyclic of same primeŽ.. sufficiently large orderwx 20 are infinite simple groups all of whose subgroups are stable. Here we initiate a program of characterizing finite groups all of whose subgroups are stable. In this paper, we establish the following two results: DEDEKIND-TYPE GROUPS 225
THEOREM 2.4. A finite sol¨able group has all its subgroups stable if and only if it is a Dedekind group.
THEOREM 3.9. A finite simple group has all its subgroups stable if and only if it is non-factorisable.
The complete list of non-factorisable simple groups can be found inwx 14 .
2. FINITE SOLVABLE GROUPS WITH ALL SUBGROUPS STABLE
The following two theorems respectively due to Baerwx 2 and Hall w 11 x will be used in this section.
THEOREM A. All subgroups of a non-abelian group G are normal if and only if G is a direct product of the quaternion group of order 8, an elementary abelian 2-group and an abelian group with all elements of odd order.
THEOREM B. A finite group G is sol¨able if and only if for e¨ery set of prime di¨isors of<< G , G has a Hall -subgroupŽ a -subgroup whose index is a X-number..
PROPOSITION 2.1. Let H be a normal subgroup of G. Then a subgroup K of G containing H is a stable subgroup of G if and only if KrH is a stable subgroup of GrH. Proof. It can be easily checked that S ª Ž.S s ÄHx< x g S4, where is the quotient map from G to GrH, is a surjective map from the set of all right transversals of K in G to the set of all right transversals of KrH in GrH such that the corresponding right quasigroups are isomorphic. The result now follows. The proof of Theorem 2.4 depends on two lemmas.
LEMMA 2.2. All subgroups of a finite p-group, where p is an odd prime, are stable if and only if it is an abelian p-group.
Proof. The proof is by induction on < By Proposition 2.1, all subgroups of M are stable. By the induction assumption M, and hence G, is abelian. Suppose, if possible, that G is non-abelian and all maximal subgroups of Gcontain all elements of order p. Let a be an element of the center ZGŽ. of G of order p. Then Gr²:a is a p-group all of whose subgroups are stableŽ. Proposition 2.1 . By the induction assumption Gr²a:is abelian. 2 Since G is assumed to be non-abelian, the commutator subgroup G s ²:a . This also shows that there is a unique subgroup of order p in ZGŽ.. Hence t 2 2 ZGŽ.is cyclic. Suppose LEMMA 2.3. Let G be a finite 2-group all of whose subgroups are stable. Then G is a Dedekind group. Proof. The proof is again by induction on < K is cyclic or generalised quaternion. Since KrL is the quaternion group of order 8, K is neither of these. Thus K cannot contain any element of order 8 and therefore K has a presentation of the type 244 ²a,b,c;asbscse,wxwxa,bsa,cse, 22 1 1 bsc⑀,cbcys by:,1Ž. 22 where ⑀ s e or a and s e or a Žobserve that b / a and c / a.. Now let us look at the elements of K of order 2. An element of K can be ␣ i j written as a b c , where ␣ s 0, 1; i, j s 0, 1, 2, 3. Consider ijij i j ji 2j bcbc sbŽ. c bcy c b2i if j is even s . ½ c2 j if j is odd 2 i 2 j Since b s e m i is even and c can never be e for j odd, the 22 elements of K of order 2 are precisely a, b and ab .ByŽ.¨ , these are the only elements of G of order 2. It is clear from the presentationŽ. 1 that these all lie in the center of K. We show that they all lie in the center of G also. The element a is already in the center of G. Let hiig H . Clearly, y1y1 ²: bhii b h g K l H is L s a . Hence bhis hb iwhere s e or a. 2 2 2 2 2 Now, b hiis bhbsbh i b s hb iishb. This shows that b and so also ab2 commutes with each element of G. Thus every element of G of order 2 lies in the center. Hence, every non-trivial subgroup of G contains a non-trivial normal subgroup. Using the induction as before, we find that all subgroups of G are normal. THEOREM 2.4. A finite sol¨able group all of whose subgroups are stable is a Dedekind group. Proof. Let G be a finite solvable group all of whose subgroups are stable. Let p be a prime dividing the order of G. By Theorem B, G has a X Hall p -subgroup H. Let P be a Sylow p-subgroup of G. Then G s PH and P l H s Ä4e . Thus H is a right transversal of P in G whose group torsion is trivial. Since P is stable, the group torsion of each right transversal of P in G is trivial and so P is normal. Thus all Sylow subgroups of G are normal and G is the direct product of its Sylow subgroups. The result now follows from Prop. 2.1, Lemma 2.2, Lemma 2.3, and Theorem A. COROLLARY 2.5. A finitely generated sol¨able group all of whose sub- groups are stable is a nilpotent group. Proof. From a result of Robinson and WehrfritzŽ for the proof, seew 24, 15.5.3, p. 459x. , a non-nilpotent finitely generated solvable group has a DEDEKIND-TYPE GROUPS 229 finite image which is not nilpotent. Since an image of a group all of whose subgroups are stable, has all its subgroups stableŽ. Proposition 2.1 and there is no finite non-nilpotent solvable group all of whose subgroups are stableŽ. Theorem 2.4 , the result follows. COROLLARY 2.6. Let G be a non-Dedekind finite group all of whose subgroups are stable and all of whose proper quotient groups are Dedekind. Then G is simple. Proof. Suppose that G is not simple. Let H be a nontrivial proper normal subgroup of G. Then GrH will be Dedekind and so G will have a normal subgroup L of index p for some prime p. Further there exists a cyclic subgroup K s ²:a of G of order a power of p such that G s K.L. Choose a right transversal S of K in G out of elements of L. Then the group torsion GS of S is not isomorphic to K. By Theorem 2.4, G is not solvable and therefore, there is a prime q / p, q ) 2 such that q divides < 3. SIMPLE GROUPS In this section, we classify finite simple groups with all subgroups stable. LEMMA 3.1. Let G be a simple group and H a subgroup of G. If S is a right trans¨ersal of H in G which generates G, then the group torsion GS of S is isomorphic to H. ²: ²: Proof. Suppose that S s G. Then HS s H l S s H. Since G is Ž. Ž. simple, the homomorphism : G s HSSSªGSgiven by hx s hx, where is the permutation representation of H on S, is an injective homomorphism. Hence GS ( H. LEMMA 3.2. Let G be a finite simple group and H a subgroup of G such that G / H.K for any proper subgroup K of G. Then H is a stable subgroup of G. Proof. Under the hypothesis of the Lemma, all right transversals of H in G will generate G. The result follows from Lemma 3.1. 230 RAMJI LAL THEOREM 3.3. Let G be a finite non-abelian simple group generated by two elements. Then all subgroups of G are stable if and only if G admits no factorisation as product of two proper subgroups. Proof. Let G be a finite non-abelian simple group generated by two elements a and b. Suppose that G s A.B, where A and B are proper subgroups of G. We can find a right transversal S of A in G from the elements of B whose group torsion GS is not isomorphic to A. Suppose that a g A. Then b f A. Since G is non-abelian simple group, there is a prime p G 5 such that p divides < COROLLARY 3.4. If G is a simple group of alternating type or a simple group of Lie type, then all its subgroups are stable if and only if it cannot be expressed as a product of two proper subgroups. Proof. Since all simple groups of alternating type and also of Lie type are 2-generator groupswx 30 , the result follows from Theorem 3.3. COROLLARY 3.5. There is no non-abelian finite simple group which has a subgroup of prime power index and all of whose subgroups are stable. Proof. This follows fromwx 1, Theorem 5.8 and Theorem 3.3 if we note that all groups inwx 1, Theorem 5.8 are factorisable and generated by two elements. COROLLARY 3.6. Let G be a finite group in which the centralizer of any in¨olution is an elementary abelian 2-group. Then G has all its subgroups stable if and only if G is an elementary abelian 2-group. DEDEKIND-TYPE GROUPS 231 Proof. From a result of Brauer et al.wx 3 , under the hypothesis of the corollary, Ž.i < THEOREM 3.7. Let G be a non-Dedekind doubly transiti¨e permutation group in which the identity is the only element fixing three elements. Then G has all its subgroups stable if and only if it is one of the following: Ž.iPSL Ž2, q ., q ' 1 Ž mod 4 . , q f Ä45, 9, 29 . 2n 1 Ž.ii Suzuki group Sz Ž.q , q s 2,q nG1. Proof. Since the Frobenius complements of a Frobenius group are non-stable it follows from Theorem 2.4 and Feitwx 7 that G has no regular normal subgroup. Thus G is a Zasenhauss group. From the classification Theorem for Zasenhauss groupsŽ seewx 12. , G is one of the following: Ž.i PSL Ž2, p r., p r) 4, 2n 1 Ž.ii Suzuki groups Sz Ž 2q. , n G 1, r r Ž.iii PGL Ž2, p ., p ) 3, p G 3, r r Ž.iv Mp Ž .,p)3, p G 3, r 2r2r where MpŽ .Žis the group of all mappings on P 1, p .Žs GF p .j Ä4ϱ of the form r ax p b q 2 r 2 x ª pr ,ifad y bc f Ž.GFŽ. p cx q d and ax b q 2 r 2 xª ,ifad y bc g Ž.GFŽ. p y Ä40. cx q d Now, from Corollary 3.4 and Itowx 13 , it follows that PSLŽ2, p r.has all its r r subgroups stable if and only if p ' 1Ž. mod 4 and p f Ä45, 9, 29 . Further, from Suzukiwx 31, Theorem 9, p. 137 and Corollary 3.4, it follows that all 232 RAMJI LAL 2 n 1 subgroups of SzŽ 2q . , n G 1 are stable. Finally, looking at the structure of the groups PGLŽ2, p r .Žand Mpr.Žseewx 12. , we find that these groups have nonstable 2-subgroups. THEOREM 3.8. Let G be a finite non-abelian simple group of order ␣1 ␣ 2 ␣ r p12p иии pr, where p12, p ,..., pri are distinct primes, ␣ G 1. Let p0s Ž. max p12, p ,..., pr and suppose that ␣23q ␣ q иии q␣rF p0y 1. Then all subgroups of G are stable if and only if it cannot be written as a product of two proper subgroups. Proof. We may assumeŽ. Corollary 3.5 that G has no subgroup of prime power index. Suppose that G s H.K, where H and K are proper 1 r subgroups of G. Suppose further, that < e, x, ax2,...,ax2q1,ax2q2,...,ax2q3q1,..., Ä 12  23213 3  иии  2 иии   1 ax2q q ry1q ,...,ax2qq ry1q rq . 1r rr 4 If T generates G, then the group torsion GT of T is isomorphic to H Ž.Lemma 3.1 . Further, we can find a right transversal T Xof H in G out of elements of K whose group torsion will not be isomorphic to H. Thus in this case H is non-stable. Next, suppose that T generates a proper 2 r subgroup L of G. Then G s H.L and p2 иии pr divides < THEOREM 3.9. All subgroups of a finite simple group are stable if and only if it is non-factorisable. 4. SOME PROBLEMS Let S be a right transversal of a subgroup H of a group G. Let H S be ²: Ž. the smallest normal subgroup of S s HSSSwhich contains H . Let MS SŽS. denote the quotient group HSSSrH (GSrH , where is as in the proof of the Lemma 3.1. Clearly, MSŽ.depends only on the right quasi- group structure on S and has a presentation ²:S; R where R s 1 Ä xyŽ. x( yxy<,ygS4is the set of relators of the presentation, ‘‘(’’ being the binary operation induced on S Ž.see the introduction . There is an obvious homomorphism ␣: S ª MSŽ.such that given any group K and a homomorphism : S ª K, there is a unique homomorphism : MSŽ.ªK such that  s ( ␣.If His a normal subgroup of G, then MSŽ.(GrH for every right transversal S of H in G. Let us call a subgroup H of a group G a prenormal subgroup of G if for any two right transversals S1 Ž. Ž. and S21of H in G, MS (MS2. The conclusions of Theorem 2.4 and Corollary 2.5 are valid if we replace stable subgroups by prenormal subgroups. It is an interesting problem to examine the results of the paper after replacing stable by prenormal. ACKNOWLEDGMENT The author is grateful to the referee for his valuable comments and suggestions for the improvement of the paper. REFERENCES 1. Z. Arad and E. Fisman, On finite factorizable groups, J. Algebra 86 Ž.1984 , 522᎐548. 2. R. Baer, Situation der Untergruppen und struktur der Gruppe, S.B. Heidelberg Akad. Mat. Nat. 2 Ž.1933 , 12᎐17. 3. R. Brauer, M. Suzuki, and G. E. Wall, A characterization of the one-dimensional unimodular projective groups over finite fields, Illinois J. Math. 2 Ž.1958 , 718᎐745. 4. B. Bruno and R. E. Phillips, Groups with restricted non-normal subgroups, Math. Z. 2 Ž.1981 , 199᎐221. 5. S. N. Cernikov,ˇ Infinite non-abelian groups in which all infinite non-abelian subgroups are invariant, Ukrain. Mat. Z. 23 Ž.1971 , 498᎐518. 234 RAMJI LAL 6. R. Dedekind, Uber Gruppen, deren samtliche¨ Teiler Normalteiler sind, Math. Ann. 48 Ž.1897 , 548᎐561. 7. W. Feit, On a class of doubly transitive permutation groups, Illinois J. Math. 4 Ž.1960 , 170᎐186. 8. L. Finkelstein, The maximal subgroups of Conway’s group Co3 and McLaughlin’s group, J.Algebra 25 Ž.1973 , 58᎐89. 9. M. S. Garascuk,ˇ Metahamiltonian linear groups, Vestnik Beloruss. Gos. Uni¨. Ser. 1 2, No. 103Ž. 1975 , 89᎐90. 10. Yu. A. Golfond, On groups all of whose subgroups are special, Dokl. Akad. Nauk. SSSR 60 Ž.1948 , 1313᎐1315. 11. P. Hall, A characteristic property of soluble groups, J. London Math. Soc. 12 Ž.1937 , 198᎐200. 12. B. Huppert and N. Blackburn, ‘‘Finite Groups III,’’ Springer, Berlin, 1982. 13. N. Ito, On factorizations of linear fractional groups LFŽ2, p n., Acta Math. Soc. Szeged 15 Ž.1953 , 79᎐83. 14. M. W. Liebeck, C. E. Praeger, and J. Saxl, The maximal factorizations of finite simple groups and their automorphism groups, Mem. Amer. Math. Soc. 86, No. 432Ž. 1990 . 15. A. A. Mahnev, Finite metahamiltonian groups, Ural. Gos. Uni¨. Mat. Zap. 10 1, No. 151 Ž.1976 , 60᎐75. 16. G. A. Miller and H. C. Moreno, Non-abelian groups in which every subgroups is abelian, Trans. Amer. Math. Soc. 4 Ž.1903 , 398᎐404. 17. V. T. Nagrebeckii, Invariant coverings of subgroups, Ural Gos. Uni¨. Mat. Zap. 5 3 Ž.1966 , 91᎐100. 18. V. T. Nagrebeckii, Finite non-nilpotent groups, any non-abelian subgroup of which is normal, Ural Gos. Uni¨. Mat. Zap. 6 80᎐88 Ž.1967 . 19. V. T. Nagrebeckii, Finite groups in which any non-nilpotent subgroups is invariant, Ural Gos. Uni¨. Mat. Zap. J. 45᎐49 Ž.1968 . 20. A. Yu. Ol’shankii, ‘‘Geometry of Defining Relations in Groups,’’ Kluwer, Dordrecht, 1989. 21. R. E. Phillips and J. S. Wilson, On certain minimal conditions for infinite groups, J. Algebra 51 Ž.1978 , 41᎐68. 22. R. Lal, Transversals in groups, J. Algebra, in press. 23. R. Lal and R. P. Shukla, Perfectly stable subgroups of finite groups, Comm. Algebra,in press. 24. D. J. S. Robinson, ‘‘A Course in the Theory of Groups,’’ Springer, Berlin, 1982. 25. G. M. Romalis and N. F. Sesekin, Metahamiltonian groups, Ural. Gos. Uni¨. Mat. Zap. 5, 3Ž.1966 , 101᎐106. 26. G. M. Romalis and N. F. Sesekin, Metahamiltonian groups II, Ural. Gos. Uni¨. Mat. Zap. 63Ž.1968 , 52᎐58. 27. G. M. Romalis and N. F. Sesekin, Metahamiltonian groups III, Ural. Gos. Uni¨. Mat. Zap. 73Ž.1969᎐1970 , 195᎐199. 28. J. E. Roseblade, On groups in which every subgroup is subnormal, J. Algebra 2 Ž.1965 , 402᎐412. 29. O. J. Schmidt, On groups every proper subgroup of which is special, Mat. Sb. Ž.N.S. 31 Ž.1924 , 366᎐372. 30. R. Steinberg, Generators for simple groups, Canad. J. Math. 14 Ž.1962 , 277᎐283. 31. M. Suzuki, On a class of doubly transitive groups, Ann. of Math. 75 Ž.1962 , 105᎐145. 32. R. A. Wilson, On maximal subgroups of the Fisher group Fi22 , Math. Proc. Cambridge Philos. Soc. 95 Ž.1984 , 197᎐222.