on locally finite minimal non-solvable groups

Cent. Eur. J. Math. • 8(2) • 2010 • 266-273DOI: 10.2478/s11533-010-0005-8

Central European Journal of Mathematics

On locally finite minimal non-solvable groups

Research Article

Ahmet Arıkan1∗, Sezgin Sezer2† Howard Smith3‡

1 Gazi Egitim Fakültesi, Matematik Egitimi Anabilim Dalı, Gazi Üniversitesi, Ankara, Turkey

2 Matematik-Bilgisayar Bölümü, Çankaya Üniversitesi, Ögretmenler Cad. No:14, Ankara, Turkey

3 Mathematics Department, Bucknell University, Lewisburg, USA

Received 30 September 2009; accepted 15 January 2010

Abstract: In the present work we consider infinite locally finite minimal non-solvable groups, and give certain character-izations. We also define generalizations of the centralizer to establish a result relevant to infinite locally finiteminimal non-solvable groups.

MSC: 20F19, 20F50, 20F14

Keywords: Minimal non-solvable groups • Locally nilpotent groups • Solvabilizer • Centralizer condition

© Versita Sp. z o.o.

1. Introduction

It was proved in [2] that a Fitting p-group in which every proper subgroup is solvable and hypercentral is itself solvable;in particular, a locally finite p-group with every proper subgroup nilpotent is solvable. For further discussion of theinteresting problem of determining the structure of such minimal non-nilpotent groups, perhaps it suffices here to referthe reader to papers [1] and [2]. In the present article we are concerned with minimal non-solvable groups, particularlyp-groups with this property; thus the paper may be regarded as a continuation of [1]. We remark at the outset that itis not known whether there exists a minimal non-solvable infinite group within the class of locally finite groups. In theprocess of our investigation we introduce (in Section 3) some generalizations of centralizers of subgroups, which mightbe of independent interest.Our main results are as follows. Throughout we use d(N) to denote the derived length of a solvable group N.

Theorem 1.1.Let G be a locally nilpotent minimal non-solvable p-group with trivial center. Then there is a positive integer t suchthat for every proper normal subgroup N of G either d(N) < t or d(CG(N)) < t .

∗ E-mail: [email protected]† E-mail: [email protected]‡ E-mail: [email protected]

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A. Arıkan, S. Sezer, H. Smith

Let G be a group and let {Xi} be a set of subsets of G. Define φ0(X1) = X1,

φr(X1, ..., X2r ) = [φr−1(X1, . . . , X2r−1 ), φr−1(X2r−1+1, . . . , X2r )]

for r ≥ 1.

Theorem 1.2.Let G be a minimal non-solvable Fitting p-group. Then G has an epimorphic image G which has a non-trivial finitesubgroup U of derived length r for a positive integer r such that

G = 〈g : φr(U, . . . , U︸︷︷︸i−1

, g, U, . . . , U︸︷︷︸2r−i

) = 1〉

for every positive integer i.

2. Proof of Theorem 1.1 and some applications

Proof of Theorem 1.1. For i ≥ 1 we define Yi = {φi(y1, . . . , y2i ) : y1, . . . , y2i ∈ G}. The first step is to provethat there exist a nontrivial finite subgroup U , a positive integer n and a proper subgroup L of G such that

⋂φn(y1,...,y2n )∈Yn\L

〈U, y1, . . . , y2n〉 6= U.

Since G is perfect, we have G = 〈Yi〉 for every i ≥ 1. Now assume that the assertion is false. Clearly G has elementsy1, y2 such that [y1, y2] 6= 1. Put U = 〈y1, y2〉, then

⋂φ1(x1,x2)∈Y1\CG ([y1,y2 ])

〈U, x1, x2〉 = U

by assumption. Let a ∈ G \ U; then there exists a non-trivial element φ1(y3, y4) ∈ Y1 \ CG([y1, y2]) such thata /∈ 〈U, y3, y4〉 and φ2(y1, y2, y3, y4) 6= 1. Suppose that we have found elements y1, . . . , y2m ∈ G such that1 6= φm(y1, . . . , y2m ) ∈ Ym and a /∈ 〈y1, . . . , y2m〉 for m > 1. Then by assumption

⋂φm(x1,...,x2m )∈Ym\CG (φm(y1,...,y2m ))

〈y1, . . . , y2m , x1, . . . , x2m〉 = 〈y1, . . . , y2m〉,

since Z (G) = 1. Then there exist elements y2m+1, . . . , y2m+1 ∈ G such that

a /∈ 〈y1, . . . , y2m , y2m+1, . . . , y2m+1 〉

and φm+1(y1, . . . , y2m , y2m+1, . . . , y2m+1 ) 6= 1. Thus we have defined inductively a sequence {y1, y2, . . . }; put X =〈y1, . . . , y2i : i ≥ 1〉. Then clearly X 6= G and φm(x1, . . . , x2m ) 6= 1 for all m ≥ 1, i.e., X is not solvable and thiscontradiction completes the proof of our assertion.Put U = {u0, u1, . . . , ur} and define

Si ={φn(y1, . . . , y2n ) : a = uib, b ∈< y1, . . . , y2n >U}

,

then Yn \ L =⋃ri=1 Si.

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If we setKi =

⋂φn(y1,...,y2n )∈Si

〈y1, . . . , y2n〉U ,

then u−1a ∈ K and so Ki 6= 〈1〉 for every i ∈ {1, . . . , r}. In particular,

Ei :=⋂

φn(y1,...,y2n )∈Si

〈y1, . . . , y2n〉G 6= 〈1〉

for every i ∈ {1, . . . , r}.Let N be a proper normal subgroup of G with d(N) > m := n + d(L). Then N(n) is not contained in L, so there existsj ≥ 1 such that N(n) ∩ Sj 6= ∅. Let φn(y1, . . . , y2n ) ∈ (N(n) ∩ Sj ). It follows that

Ej ≤ 〈y1, . . . , y2n〉G ≤ N

and hence CG(N) ≤ CG(Ej ). Put W = 〈CG(N) : N C G and d(N) > m〉; then we have

W ≤ CG(E1) . . . CG(Er)

which is a proper subgroup of G, since each CG(Ei) is normal and solvable (as Z (G) = 1). Therefore we have W 6= G.Put t = max{d(CG(E1)) . . . , d(CG(Er)), m}. If N is a proper normal subgroup of G such that d(N) > m, then CG(N) ≤CG(Ei) for some i and hence d(CG(N)) ≤ t, while if d(N) ≤ m, then clearly d(N) ≤ t. So the theorem is proved.

The positive integer t defined in the proof of Theorem 1.1 will be used in the rest of the paper without further reference.Now we can give some applications of Theorem 1.1.

Corollary 2.1.Let G be a locally nilpotent minimal non-solvable p-group with trivial center and let N,M be two proper normalsubgroups of G such that d(N) > t and d(M) > t. Then [N,M] 6= 〈1〉 and hence N ∩M 6= 1.

Proof. If [N,M] = 〈1〉, then N ≤ CG(M), but this contradicts Theorem 1.1. Since [N,M] ≤ N ∩ M, the resultfollows.

Let G be a group with a proper subgroup N. If CG(N) � N, then we say that N satisfies the centralizer condition.

Theorem 2.1.Let G be a locally nilpotent p-group with all proper subgroups solvable. If every proper normal subgroup of G satisfiesthe centralizer condition, then G is solvable.

Proof. Let r be a positive integer with r > t and N be a normal subgroup of G maximal such that d(N) = r. Nowlet y ∈ CG(N) \N; then [〈yG〉, N] = 〈1〉. By Theorem 1.1, d(〈yG〉) < t, and so we have d(N〈yG〉) = d(N), contradictingthe maximality of N.

Theorem 2.2.Let G be a locally nilpotent minimal non-solvable p-group with trivial center. If 〈CG(N) : N C G〉 = G, then

〈CG(N) : N C G and d(N) ≤ t〉 = G.

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Proof. We have

G = (〈CG(N) : N C G and d(N) ≤ t〉).(〈CG(N) : N C G and d(N) > t〉).

As in the proof of Theorem 1.1, we see that

G = 〈CG(N) : N C G and d(N) ≤ t〉.

Theorem 2.3.Let G be a locally nilpotent minimal non-solvable p-group with trivial center. If CG(N) 6= 1 for every proper normalsubgroup N of G, then G has a proper non-trivial normal subgroup which satisfies the centralizer condition.

Proof. Assuming the assertion false, we have that

G = 〈CG(N) : N C G, d(N) < t〉 = 〈Z (N) : N C G, d(N) < t〉 = 〈A C G : A is abelian 〉.

However, with U and L as in the proof of Theorem 1.1, we may apply Lemma 2.1 of [1] to obtain a contradiction.

Let G be a Fitting group and y be a non-trivial element of G. Then 〈yG〉 is nilpotent and hence has non-trivialcenter. So let 1 6= x ∈ Z (〈yG〉) and thus 〈yG〉 ≤ CG(〈xG〉), i.e. y ∈ CG(〈xG〉). Now clearly G = 〈CG(N) :N is an abelian normal subgroup of G〉. So Fitting groups already satisfies the conclusion of Theorem 2.2. Clearlyevery perfect locally nilpotent minimal non-hypercentral group also has this property.Again with our earlier notation, since W < G it follows that

R1 := 〈Z (N) : N C G and d(N) > m〉

is a proper subgroup of G. This generalizes as follows.

Lemma 2.1.Let G be a locally nilpotent minimal non-solvable p-group with trivial center. Then for every positive integer k , thereexists a positive natural number mk such that Rk = 〈Zk (N) : N / G, d(N) > mk〉 6= G.

Proof. We see that the assertion is true for k = 1. Assume that Rk−1 6= G for some k > 1 and put Z (G/Rk−1) =Z/Rk−1, then Z (G/Z ) = 1, since G is perfect, and by the case k = 1 there exists a natural number mk such thatG/Z 6= 〈Z (N/Z ) : Z ≤ N and N / G, d(N) > mk〉 ≥ 〈Zk (N)Z/Z : N / G, d(N) > mk〉. This implies that Rk 6= G.

3. Generalizations of the Centralizer

Define ψ0(X1) = X1,ψr(X1, . . . , Xr) = [X1, . . . , Xr ]

for r ≥ 1.The following definition includes certain generalizations of the centralizer and it seems that these are useful tools forstudying groups.

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Definition 3.1.Let G be a group and U be a subgroup of G. For r ≥ 1 and 1 ≤ i ≤ r we define

C r,iG (U) = {g ∈ G : ψr(U, . . . , U︸︷︷︸

i−1

, g, U, . . . , U︸︷︷︸r−i

) = 1}

andSr,iG (U) = {g ∈ G : φr(U, . . . , U︸︷︷︸

i−1

, g, U, . . . , U︸︷︷︸2r−i

) = 1}.

If U is solvable, then we define the solvabilizer of U in G as

SzG(U) = {g ∈ G : 〈U, g〉 is solvable and d(〈U, g〉) = d(U)} (c.f. Section 5 of [1])

and if U is nilpotent, then we define the nilpotentizer of U in G

NzG(U) = {g ∈ G : 〈U, g〉 is nilpotent and n(〈U, g〉) = n(U)}

where n(K ) denotes the nilpotency class of a nilpotent subgroup K of G.

Actually we do not need C r,iG (U) and NzG(U) in the sequel but we give the definitions for completeness and to present

them to the reader.C r,iG (U), Sr,iG (U) and SzG(U) may not be subgroups. We also have that

C 1,iG (U) ⊆ C 2,i

G (U) ⊆ · · · ⊆ C j,iG (U) ⊆ . . .

and thatS1,iG (U) ⊆ S2,i

G (U) ⊆ · · · ⊆ Sj,iG (U) ⊆ . . . .

Next we observe that Zi−1(U) ⊆ C i,1G (U) for every i ≥ 1 and if U is nilpotent of class c, then U ⊆ C c+1,1

G (U). If U = G,then C i,1

G (G) = Zi−1(G) for i ≥ 1 and in this case {C i,1G (G)} is the upper central series of G.

If N is a normal subgroup of G, then it is routine to show that C r,iG (N) and Sr,iG (N) are normal subgroups of G. But

SzG(N) may not be a subgroup:

Example 3.1.Let p be an odd prime and let U be a non-abelian group of order p3 and exponent p. Thus U is generated by elements

x and y with [y, x] = z, say. Let[a bc d

]∈ X := SL(2, p). Then the assignment x 7→ xaybzab/2, y 7→ xcydzcd/2

determines an automorphism σα of U , and the exponents of z here are precisely those that ensure that σα commutes withthe automorphism τ of U that maps x to x−1, y 7→ y−1 (and z 7→ z). If α1, α2 ∈ X , then σα1σα2 and σα1α2 certainly agreein their action on x and y modulo U ′, while the fact that each of σα1 , σα2 commutes with τ ensures that their productcommutes with τ and hence that the exponents on z agree as they should. Thus we have a homomorphism from X toAut U , and we form the natural split extension G of U by X .

Setting α =[

1 10 1

], β =

[1 01 1

], we have that each of 〈U, α〉, 〈U, β〉 is metabelian and hence that α, β ∈ SG(U). But

〈α, β〉 = X while τ /∈ SG(U), since [U, 〈τ〉] = U and hence 〈U, τ〉 has derived length three. A similar example may beconstructed for p = 2 if we begin with a group U that is a copy of Q8, the quaternion group of order 8.

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Let N be a normal subgroup of a group G. Define for n ∈ Z+ ∪ {0},

Tn,n(N) = CG(N(n)), Tn,n−1(N) = CG(N(n−1)Tn,n(N)/Tn,n(N)), . . . ,

Tn,2(N) = CG(N ′′Tn,3(N)/Tn,3(N)), Tn,1(N) = CG(N ′Tn,2(N)/Tn,2(N)),

Tn,0(N) = CG(NTn,1(N)/Tn,1(N)).

The proof of the following is straightforward and is omitted here.

Lemma 3.1.Let G be a group and let N be a normal subgroup of G. Then, for every i ≥ 1, we have

(i) C i+1,1G (N) = CG(NC i,1

G (N))/C i,1G (N))

and

(ii) Si+1,1G (N) = Ti,0(N).

We also have thatT0,0(N) ≤ T1,0(N) · · · ≤ Tn,0(N) ≤ . . . .

In particular, φn+1(G,2n+1−1 N) = [G,N,N ′, . . . , N(n−1), N(n)] = 1 if and only if G = Tn,0(N) for some n.

(iii) G is solvable if and only if Sn,1G (G) = G for some n ≥ 1.

As we see, Si,1G (G) behaves like an “upper derived series” of G, but G might be solvable with trivial center and so thefirst term S1,1

G (G) = Z (G) might be trivial.

One may ask whether φn+1(x,2n+1−1 U) = [x, U,U ′, . . . , U (n)] = 1 always implies 〈x, U〉(n) = 1? The following exampleshows that the answer is no.

Example 3.2.Let Q = Q8; then Aut Q has order 24. (Note that Q is not contained in this group since Z (Q) 6= 1.) Choose z of order3 in Aut Q and form the semidirect product H := Q o 〈z〉, we have Q C H, which has order 24, derived length 3 andQ′ ≤ Z (Q), and hence [H,Q,Q′] = 1. But H = 〈z, Q〉 and H ′′ 6= 1.

Now we embark on the proof of Theorem 1.2 which is an application of generalized centralizers.

Proof of Theorem 1.2. By [1, Theorem 2.6] G has a non-trivial homomorphic image G with Z (G) = 1 such thatfor every proper subgroup L and for every finite proper subgroup U

⋂y∈G\L

〈U, y〉 = U.

By [1, Lemma 5.1], G contains a finite subgroup V such that 〈SzG(V )〉 = G. We also have that there is a positive integerr such that 〈SzG(V )〉 ≤ 〈Sr,iG (V )〉 where r = d(V ) and hence 〈Sr,iG (V )〉 = G for every positive integer i and the resultfollows.

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4. Fitting subgroups

In this final section, we collect some further results that may prove useful in the investigation of locally finite p-groupsthat are minimal non-solvable. We remark that it is easy to show that every locally nilpotent minimal non-solvablegroup is a p-group, but it does not appear to be known whether an infinite locally finite minimal non-solvable group isa p-group.Define F1(G) = Fit(G), Fα (G) =

⋃β<α Fβ(G) if α is a limit ordinal and

Fα (G)/Fα−1(G) = Fit(G/Fα−1(G))

if α is not a limit ordinal. If it is clear what the group is, then we use Fα in place of Fα (G).

Lemma 4.1.Let G be a locally nilpotent minimal non-solvable p-group, then G = Fω.

Proof. Let N be a proper normal subgroup of G, then N is solvable of derived length d, say, and clearly N ≤ Fd.Since G is generated by proper normal subgroups, the result follows.

Lemma 4.2.Let G be a locally nilpotent minimal non-solvable p-group, then G has a series

1 E T1 E T2 · · · E Ti E . . .

such that G = Tω and Z (G/Tj ) = 1 for every positive integer j .

Proof. For each i ≥ 1, let Z (G/Fi) = Ti/Fi. Since G is perfect, we have Z (G/Ti) = 1 for all i, and it is easy to seethat Ti ≤ Ti+1 for all i. The result follows from Lemma 4.1.

Lemma 4.3.Let G be a locally nilpotent minimal non-solvable p-group, then CG(Fi) = Z (Fi) for all i ≥ 1. In particular,

Z (F1) ≥ Z (F2) ≥ · · · ≥ Z (Fi) ≥ . . . .

Proof. If G = Fi for some i ≥ 1, then the result follows for i. So we may assume that G 6= Fi for all i ≥ 1.Since every proper normal subgroup of G is solvable, F ∩N 6= 1 for every non-trivial normal subgroup N of G. Assumethat CG(F1) 6= Z (F1) and put F = F1 and C = CG(F ). Now we follow the argument in the proof of [3, 5.4.4 (ii)]. Thereis an abelian group A/F such that F < A ≤ CA. Since A = A ∩ CF = F (A ∩ C ) and γ3(A ∩ C ) ≤ [A′, C ] ≤ [F, C ] = 1,A ∩ C ≤ F and hence A = F , a contradiction. Thus CG(F1) = Z (F1) ≤ F1.By definition Fit(G/Fi) = Fi+1/Fi. Now we have that

CG(Fi+1)Fi/Fi ≤ CG/Fi (Fi+1/Fi) ≤ Fi+1/Fi

by the first paragraph and hence CG(Fi+1) ≤ Fi+1. This implies that CG(Fi+1) = Z (Fi+1). In addition, we see that

Z (F1) ≥ Z (F2) ≥ · · · ≥ Z (Fi) ≥ . . . .

So the proof is complete.

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Lemma 4.4.Let G be a locally nilpotent minimal non-(solvable and hypercentral) p-group. Then there is a bound on the hypercentrallength of proper normal subgroups of G.

Proof. By Theorem 1.1 of [2] Fi 6= G for all i ≥ 1. Let βi be the hypercentral length of Fi for each i ≥ 1 and putβ =

⋃∞i=1 βi. Let N be a proper normal subgroup G; then, as in the proof of Lemma 4.1, there is a positive integer k

such that N ≤ Fk . Hence h(N) ≤ h(Fk ) = βk ≤ β. So β is the desired bound.

Lemma 4.5.Let G be a locally nilpotent minimal non-solvable p-group. If every proper subgroup of G is hypercentral, then Zωn(N) ≤Fn for every positive integer n and proper normal subgroup N of G.

References

[1] Arıkan A., Characterizations of minimal non-solvable Fitting p-groups, J. Group Theory, 11(1), 95–103, 2008[2] Asar A.O., Locally nilpotent p-groups whose proper subgroups are hypercentral or nilpotent-by-Chernikov, J. London

Math. Soc. (2), 61(2), 412–422, 2000[3] Robinson D.J.S., A course in the theory of groups, Springer-Verlag, New York, Heidelberg, Berlin, 1982

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on locally finite minimal non-solvable groups

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