aa r X i v : . [ m a t h . G R ] M a y Finite Just Non-Dedekind Groups
V.K. Jain ∗ and R.P. ShuklaDepartment of Mathematics, University of AllahabadAllahabad (India) 211 002 Email: [email protected]; [email protected]
Running Title:
Finite Just Non-Dedekind Groups.
Abstract:
A group is just non-Dedekind (JND) if it is not a Dedekind groupbut all of whose proper homomorphic images are Dedekind groups. The aimof the paper is to classify finite JND-groups. ∗ supported by UGC, Government of India ey-words: Monolith, Characteristically simple, Semisimple.
MSC 2000:
A group is called Dedekind if all its subgroups are normal. By [1], a groupis Dedekind if and only if it is abelian or the direct product of a quaterniongroup of order 8, an elementary abelian 2-group and an abelian group withall its elements of odd order (one can also see its proof in [10, 5.3.7, p.143]).Given a group theoretical property P , a just non P -group is a group whichis not P -group but all of whose proper homomorphic images are P -groups;for brevity we shall call these JN P -groups. M.F. Newman studied just non-abelian (JNA) groups in [8, 9]. S. Franciosi and others studied solvable justnonnilpotent (JNN) groups in [3] and D.J.S. Robinson studied solvable justnon-T (JNT) groups in [11](here the group with property T means in thegroup normality is a transitive relation).The aim of this paper is to classify finite JND-groups. In Section 2, weprove that JND-groups are monolithic group. Section 3 deals with solvableJND-groups and Section 4 shows that nonsolvable JND-groups are semisim-ple. Theorem 4.4 gives complete classification of finite semisimple JND-groups.Let G be a group. For, sets X , Y of G , let [ X, Y ] denote the subgroup of G generated by [ x, y ] = xyx − y − , x ∈ X, y ∈ Y . The derived series of G is G = G (0) ≥ G (1) ≥ · · · ≥ G ( n ) ≥ · · · , where G ( n ) = [ G ( n − , G ( n − ], the commutator subgroup of G ( n − . The lowercentral series of G is G = γ ( G ) ≥ γ ( G ) ≥ · · · ≥ γ n ( G ) ≥ · · · , where γ n +1 ( G ) = [ γ n ( G ) , G ]. The group G is called solvable of derived length n (respectively nilpotent of class n ) if n is the smallest nonnegative integersuch that G ( n ) = { } (respectively γ n +1 ( G ) = { } ). We recall that a group is called monolithic if it has smallest nontrivial normalsubgroup, called the monolith of G . In this section, we study some basic2roperties of JND-groups. Proposition 2.1.
Let G be a JND-group. Then G is not contained in adirect product of Dedekind groups.Proof. Let { H i } i ∈ I denote a family of Dedekind groups, where I is an index-ing set. Assume that G is contained in H = Q j ∈ I H j . Since G is nonabelian,there exists i ∈ I such that H i is nonabelian. By the classification Theoremfor nonabelian Dedekind groups [10, 5.3.7, p.143], square of each elementof a nonabelian Dedekind group is central and its commutator subgroup isisomorphic to Z . This implies that G can not be simple. Take any non-trivial element x ∈ G . Then x ∈ Z ( G ) if x = 1 and x ∈ Z ( G ) if x = 1(for G is contained in H ). This proves that each subgroup of G contains anontrivial central element of H . Let N be a nontrivial subgroup of G . Let x ∈ N ∩ Z ( H ), x = 1. Since G is JND, G/ h x i is Dedekind, so N/ h x i E G/ h x i ,which proves that N E G . Hence G is Dedekind. Corollary 2.2.
Let G be a JND-group. Then G is monolithic.Proof. If G is a JNA-group, there is nothing to prove for G (1) will be con-tained in each nontrival normal subgroup of G . Assume that G is not JNA.Let A denote the set of all nontrivial normal subgroups of G . Then G/H isDedekind for all H ∈ A . Further, since G is not JNA, there exists H ∈ A such that G/H is nonabelian. Therefore by Proposition 2.1, the homomor-phism from G to Q H ∈A G/H which sends x ∈ G to ( xH ) H ∈A is not one-one.This proves that T H ∈A H = { } . Corollary 2.3.
Let G be as in Corollary 2.2. Assume that G (2) = { } . Thenthe monolith of G is G (2) .Proof. By Corollary 2.2, G is monolithic. Let K denote the monolith of G .Then K ⊆ G (2) . If G is JNA, then K = G (1) and so K = G (2) . If G is JNDbut not JNA, then G/K is nonabelian Dedekind. Now by [10, 5.3.7, p.143],the commutator subgroup G (1) /K of G/K is of order 2. So G (2) ⊆ K .3 Finite solvable JND-groups
In this section, we classify finite solvable JND-groups. Solvable JNA-groupswith nontrival center is characterized in [9] and centerless solvable JNA-groups have been classified in [8, Theorem 5.2, p.360]. So, it only remains toclassify finite solvable JND-groups which are not JNA-groups.
Proposition 3.1.
Let G be a JND-group. Let Z ( G ) , the center of G benontrivial. Then G is a solvable JNA-group.Proof. Suppose that G is JND but not JNA. By Corollary 2.2, G is mono-lithic. Let K denote the monolith of G . Since every subgroup of Z ( G ) isnormal subgroup of G , K is central subgroup of order p for some prime p .We claim that p = 2. Since G is JND but not JNA, G/K is nonabelianDedekind. By the structure Theorem for nonabelian Dedekind groups [10,5.3.7, p.143], the commutator (
G/K ) (1) = G (1) /K is of order 2. Thus | G (1) | =2 p . Let x be an element of G (1) of order 2. If x ∈ Z ( G ), then K = h x i ,so p = 2. Assume that x Z ( G ). Since | G (1) /K | = 2 and x K , so G (1) = h x i K . Let g ∈ G such that gxg − = x . Then there exists a nontrivialelement h ∈ K such that gxg − = xh . Now since h ∈ Z ( G ), h = x h =( xh ) = ( gxg − ) = 1 implies that p = 2.Next, we show that G does not contain an element of odd prime order.Assume that x ∈ G is of odd prime order q . Since h x i K has a uniquesubgroup of order q and h x i K E G (for G/K is Dedekind), h x i E G . But,then K ⊆ h x i , a contradiction.Further, since G/K is a nonabelian Dedekind, by [10, 5.3.7, p.143], G does not contain any element of infinite order. Thus we have shown that G is a 2-group. Lastly, since G/K is a nonabelian Dedekind, by [10, 5.3.7,p.143], G contains a nonabelian subgroup H of order 16 such that K ⊆ H and H/K ∼ = Q . But this is not possible [2, 118, p.146]. Lemma 3.2.
A finite centerless solvable JND-group is a JNT-group.Proof.
Let G be a finite centerless solvable JND-group. Since a Dedekindgroup is also a T-group, it is sufficient to show that G is not a T -group.Suppose that G is a T -group. Let K denote the monolith of G (Corollary2.2). Since G is a finite solvable T -group, K is a cyclic group of order p p . Since G/K is nonabelian Dedekind group, by [10, 5.3.7,p.143] , | G (1) /K | = 2. Further, since a solvable T -group is of derived lengthat most two [10, 13.4.2, p.403], G (1) is abelian. Now since G (1) is an abeliangroup of order 2 p and G is a T -group, p = 2. But then K ⊆ Z ( G ) = { } .This is a contradiction. Therefore G is a JNT-group.The following example shows that there exists a solvable JND-groupwhich is not a JNA-group. Example 3.3.
Consider an elementary abelian 3-group A of order 9. Let ψ denote the homomorphism from Q to Aut A = Gl (3) defined as ı (cid:18) − (cid:19) , (cid:18) − (cid:19) , where Q = {± , ± ı, ± , ± k } is thequaternion group of order 8. It is easy to check that ψ is injective. Let G = AQ denote the natural semidirect product of A by Q . Then G is aJND-group with monolith A .The following proposition classifies all finite solvable JND-groups whichare not JNA-groups. Lemma 3.4.
A finite solvable group G is JND but not JNA if and only ifthere exists an elementary abelian normal p -subgroup A of G for some prime p which is also monolith of G and a nonabelian Dedekind group X of G suchthat A ∩ X = { } , G = AX and the conjugation action of X on A is faithfuland irreducible.Proof. Suppose that G is a finite solvable JND-group but not JNA-group.By Corollary 2.2, G is monolithic. Let K be the monolith of G . Then G/K is a nonabelian Dedekind. Thus by [10, 5.3.7, p.143], | G (1) /K | = 2. Since K is characteristically simple and abelian, it is an elementary abelian p -groupof order p n for some prime p [10, 3.3.15 (ii), p.87].Assume that G (1) is abelian. If p = 2, then G (1) contain unique elementof order 2 and so Z ( G ) = 1. By Proposition 3.1, this is a contradiction. Thus p = 2. Now by Proposition 3.1, G is not nilpotent and by Lemma 3.2, G isa JNT-group. So by Case 6.2 and its Subcases 6.211, 6.212, 6.22 and 6.222in [11, pp.202-208], there is no finite JNT-group which is not JNA and has aminimal normal subgroup isomorphic to an elementary abelian 2-group.5ssume that G (1) is nonabelian. Then [ G (1) , K ] = 1, for | G (1) /K | = 2.Now since G is a finite nonnilpotent JNT-group and [ G (1) , K ] = 1, by Case6.1 of [11, p.202], there is a nontrival normal subgroup A of G , a solvable T -subgroup X of G such that A ∩ X = { } , G = AX and the conjugationaction of X on A is faithful and irreducible. Further, since K ⊆ A and theconjugation action of X on A is irreducible, K = A . So X ∼ = G/A = G/K isa nonabelian Dedekind group.Conversely, suppose that G = AX , A ∩ X = { } , X is a nonabelianDedekind subgroup, A is an elementary abelian p -group and also the monolithof G . Since A is solvable and G/A ∼ = X is nonabelian Dedekind and sosolvable, G is solvable. Further, since A is the monolith of G and G/A ∼ = X is nonabelian Dedekind, G is JND but not JNA.The following proposition lists some more properties of finite solvableJND-groups which are not JNA-groups. Proposition 3.5.
Let G , A and X be as in the Lemma 3.4. Then(i) The stabilizer of any nontrival element of A is trival.(ii) | X | divides p n − , in particular p and | X | are coprime.(iii) X ∼ = Q × A o , where A o is a cyclic group of odd order.Proof. Let a ∈ A, a = 1. Assume that the stabilizer stab X ( a ) of a in X is nontrival. Assume that x ∈ stab X ( a ) , x = 1. Since G/A is a Dedekindgroup, h x i A E G . Thus Z ( h x i A ) is a nontrival normal subgroup of G (for a ∈ Z ( h x i A )) and so A ⊆ Z ( h x i A ), for A is the monolith of G . But this isa conradiction, for the conjugation action of X on A is faithful. This proves(i). Now (ii) is implied by the class equation for the action of X on A .Further, by [11, Lemma 1, p.185], there is an extension field E of Z p suchthat Z ( X ) ∼ = Y ≤ E ⋆ and E = Z p ( Y ), where E ⋆ denote the multiplicativegroup of E . Clearly E is a finite field , so E ⋆ is a cyclic group. This implies X ∼ = Q × A o , where A o is a cyclic group of odd order [10, 5.3.7, p.143]. Thisproves (iii). 6 Finite nonsolvable JND-groups
Recall that a group is semisimple [10, p.89] if its maximal solvable normalsubgroup is trivial. Also a maximal normal completely reducible subgroupis called the
CR-radical [10, p.89].
Proposition 4.1.
Let G be a finite nonsolvable JND-group. Then G is asemisimple group.Proof. Assume that G has a nontrival normal solvable subgroup N . Then G/N is a Dedekind group. Hence by [10, 5.3.7, p.143],
G/N is solvable. Butthen G is solvable, a contradiction.Now we fix some notations for the rest of the section. For a group G , wedenote Inn G for the inner automorphism subgroup of Aut G , the automor-phism group of G and Out G for the outer automorphism group of G . Let H denote a finite nonabelian simple group. Consider the semidirect product( Aut H × . . . × Aut H ) | {z } r copies ⋊ S r and ( Out H × . . . × Out H ) | {z } r copies ⋊ S r , where S r actson ( Aut H × . . . × Aut H ) | {z } r copies as well as on ( Out H × . . . × Out H ) | {z } r copies by permut-ing the coordinates. Let˜ ν : ( Aut H × . . . × Aut H ) | {z } r copies ⋊ S r −→ ( Out H × . . . × Out H ) | {z } r copies ⋊ S r be the homomorphism defined by ˜ ν ( x , x , . . . , x r , x r +1 ) = ( x Inn H, . . . ,x r Inn H, x r +1 ). We denote by β the projection of ( Out H × . . . × Out H ) | {z } r copies ⋊ S r onto the ( r + 1)-th factor S r , which is obviously a homomorphism. Lemma 4.2.
Let H be a finite nonabelian simple group. Then Out H doesnot contain a subgroup isomorphic to the quaternion group Q of order .Proof. If H is isomorphic to either alternating group Alt n of degree n or toa Sporadic simple group, then | Out ( H ) | ≤ H is isomorphic7o a finite simple group of Lie type, then the Lemma follows by [4, Theorem2.5.12, p.58]. Corollary 4.3.
Let H be a finite nonabelian simple group. Then for any m ∈ N , Out H × . . . × Out H | {z } m copies does not contain a subgroup isomorphic to thequaternion group Q of order .Proof. Assume that α is an injective homomorphism from Q to Out H × . . . × Out H | {z } m copies . Let u = ( x , x , . . . , x m ) denote an element of α ( Q )of order 4. Then there is t (1 ≤ t ≤ m ) such that x t is of order 4. Let p t denote the projection of Out H × . . . × Out H | {z } m copies onto the t -th factor. Then( p t ◦ α )( Q ) is a subgroup of Out H which contains an element of order4. Since a homomorphic image of Q containing an element of order 4 isisomorphic to Q , ( p t ◦ α )( Q ) ∼ = Q . By Lemma 4.2, this is impossible. Theorem 4.4.
A finite nonsolvable group G is JND-group if and only ifthere exists a finite nonabelian simple group H , a natural number r and aDedekind group D ⊆ ( Out H × . . . × Out H ) | {z } r copies ⋊ S r such that(i) the usual action of β ( D ) on the set { , , . . . , r } is free and transitive,and(ii) G ∼ = ˜ ν − ( D ) ,where all the notations have meaning described as after the Proposition 4.1Further, G is JND but not JNA if and only if D is a nonabelian Dedekindgroup and r is even.Proof. Suppose that G is a nonsolvable JND-group. By Corollary 2.2, G is monolithic. Let K denote the monolith of G . Since G is nonsolvableand K is characteristically simple, by [10, 3.3.15 (ii), p.87], there exists a8nite nonabelian simple group H and a natural number r such that K ∼ =( H × . . . × H ) | {z } r copies .By Proposition 4.1, G is semisimple. We show that K is the CR-radicalof G . Let N be the CR-radical of G containing K . Then N is semisimple [5,Lemma, p.205]. Assume that N = K . Then there exists nontrival completelyreducible normal subgroup L of N which is complement of K in N . Nowsince L ∼ = N/K and
G/K is a Dedekind group, L is solvable [10, 5.3.7, p.143].Further, since nontrival normal subgroup of a semisimple group is semisimple[5, Lemma, p.205], L is also semisimple. This is a contradiction.Now by [10, 3.3.18 (i), p.89], there exists G ⋆ ∼ = G such that( Inn H × . . . × Inn H ) | {z } r copies ≤ G ⋆ ≤ ( Aut H × . . . × Aut H ) | {z } r copies ⋊ S r . We identify G with G ⋆ and H with Inn H . Thus K is identified with ( Inn H × . . . × Inn H ) | {z } r copies .Take D = G/K ⊆ ( Out H × . . . × Out H ) | {z } r copies ⋊ S r . Then D is a Dedekind groupand G ∼ = ˜ ν − ( D ). This proves (ii).Next, we claim that β ( D ) acts transitively on the set of symbols { , , . . . ,r } . Let O denote an orbit of the natural action of β ( D ) on { , , . . . , r } .Consider the subgroup M O = { ( x , x , . . . , x r , | x i ∈ Inn H and x i = 1 if i O } of ( Aut H × . . . × Aut H ) | {z } r copies ⋊ S r . It is easy to observe that for each elementof G , the ( r + 1)-th coordinate is an element of β ( D ). This implies that M O is a normal subgroup of G contained in K . But K is the monolith of G , so M O = K . This proves that O = { , , . . . , r } .Now, we show that the action of Z ( β ( D )) on { , , . . . , r } is free. Supposethat an element u of Z ( β ( D )) fixes a symbol a under the natural action of β ( D ) on { , , . . . , r } . Clearly u fixes each element of the orbit β ( D ) .a of a which is { , , . . . , r } . This implies u = 1. So, no nontrivial element of Z ( β ( D )) will fix any symbol in { , , . . . , r } .If D is abelian, then Z ( β ( D )) = β ( D ) and so the action of β ( D ) isfree. If D is nonabelian Dedekind group, then by the structure Theorem forDedekind groups [10, 5.3.7, p.143], there exists a nonnegative integer t and anabelian group A o of odd order such that we can write D = Q × ( Z ) t × A o .Thus for any x ∈ β ( D ), either x ∈ Z ( β ( D )) or 1 = x ∈ Z ( β ( D )). Thisimplies that a noncentral element x also does not fix any symbol of set9 , , . . . , r } (for then 1 = x ∈ Z ( β ( D )) will fix that symbol). Thus actionof β ( D ) on { , , . . . , r } is free and transitive. In particular r = | β ( D ) | . Thisproves (i).Now, assume that G is JND but not JNA. Then by Corollary 4.3, β ( Q ) =1 and so 2 divides | β ( D ) | = r Conversely, suppose that there exists a Dedekind group D ⊆ ( Out H × . . . × Out H ) | {z } r copies ⋊ S r for some r ∈ N and a nonabelian finite sim-ple group H such that, the usual action of β ( D ) on the set { , , . . . , r } is free and transitive. Let G = ˜ ν − ( D ). We will show that G is a JND-group. By [10, 3.3.18 (ii), p.89], G is semisimple with CR-radical K = ( Inn H × . . . × Inn H ) | {z } r copies and ( Inn H × . . . × Inn H ) | {z } r copies ≤ G ≤ ( Aut H × . . . × Aut H ) | {z } r copies ⋊ S r . We will show that K is the monolith of G .Since K = K (1) , K is contained in all terms of the derived series of G .Further, since G is semisimple, there is smallest nonnegative integer n suchthat G ( n ) = G ( n + i ) for all i ∈ N . This implies that G ( n ) /K is a perfectgroup. But since G ( n ) /K is Dedekind and so solvable [10, 5.3.7, p.143], G ( n ) = K . Let N be a nontrival normal subgroup of G . Since a nontrivialnormal subgroup of a semisimple group is semisimple [5, Lemma, p.205] anda semisimple group has trivial center, N ∩ K = { } . By [7, Theorem 2, p.156], N ∩ K = N × N × . . . × N r | {z } r copies , where N i E Inn H and at least one N i = 1.Now since N i = Inn H and β ( D ) acts transitively on Inn H × . . . × Inn H | {z } r copies ,so N ∩ K = Inn H × . . . × Inn H | {z } r copies = K , that is K ⊆ N . This proves that K is the monolith of G . Thus G is JND-group. Further, if D is nonabelianDedekind group, then G is JND but not JNA. Remark . Let G be finite just nonsolvable (JNS) (respectively just non-nilpotent (JNN)) group. Let n be the smallest nonnegative integer such that G ( n ) = G ( n + k ) (respectively γ n ( G ) = γ ( n + k ) ( G )) for all k ∈ N . Then it is easyto see that G ( n ) (respectively γ ( n +1) ( G )) is the monolith of G .10he idea of the proof of the above theorem can be used to show that:A finite nonsolvable group G is JNS-group (respectively JNN-group) if andonly if there exists a finite nonabelian simple group H , a natural number r and a solvable (respectively nilpotent) group D ⊆ ( Out H × . . . × Out H ) | {z } r copies ⋊ S r such that(i) the usual action of β ( D ) on the set { , , . . . , r } is transitive,and(ii) G ∼ = ˜ ν − ( D ),where all the notations have meaning described as after the Proposition 4.1. Acknowledgement:
We thank Professor Ramji Lal for suggesting the prob-lem and for several stimulating discussions.
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