On the group of automorphisms of finite wreath products

On the Group of Automorphisms of Finite Wreath Products

FRANCESCOFUMAGALLI(*)

Dedicated to Guido Zappa on his 90th birthday ABSTRACT- Inthis paper we study the structure of the group of automorphisms of the wreath productA wr C2, whereAis a finite nilpotent group, andC2is the cyclic group of order2. In particular we give necessary and sufficient conditions for Aut (A wr C2) to be supersolvable depending only on Aut (A) and on the Remak decompositionofA.

Introduction and statement of the main result.

This work is a contribution to the study of the group of automorphisms of a (restricted) wreath productA wr Bof two non-trivial groupsAandB.

Throughout this paper we denote with G the group A wr B. Particular interest concerns the relationships among the structures and the group- theoretical properties of the groupsAandBand the ones of Aut (G).

For instance, if Aut (G) is a nilpotent group much is known. In fact in this caseGis clearly nilpotent and for a result of G. Baumslag ([2]) we have that bothAandBare nilpotentp-groups (for the same primep), whereA has finite exponent andBhas finite order. Moreover, if Aut (G) is supposed to be finite (and nilpotent), then Aut (A) and Aut (B) are finite p-groups (see [9]). Conversely, ifAandBare finitep-groups (forp6ˆ2) with Aut (A) and Aut (B)p-groups too, thenAut (A wr B) is ap-group (see M.V. Khor- oshevskii [4]).

Concerning supersolvability, in a recent paper ([7]) G. Corsi Tani and R. Brandl proved the following

(*) Indirizzo dell'A.: UniversitaÁ degli Studi di Firenze, Dipartimento di Matematica ``Ulisse Dini'', viale Morgagni 67A, 50134 Firenze, Italia; e-mail:

fumagalli@math.unifi.it

A work partially supported by MURST research program ``Teoria dei gruppi e applicazioni''.

THEOREM. IfAut (G)is supersolvable then A is nilpotent. Moreover if either G is supposed to be infinite or if(jAj;jBj)ˆ1, then B'C2;the cyclic group of order two.

In this paper we study the ``inverse problem'' of the aforementioned result. We suppose thatAis a finite nilpotent group and give necessary and sufficient conditions for Aut (A wr C2) to be supersolvable.

Our mainTheorem is proved insection3 and it canbe stated as follows.

THEOREM 1. Let A be a finite nilpotent group with Aut (A) super- solvable and let G:ˆA wr C₂. ThenAut (G)is supersolvable if and only if either no two distinct factors in a Remak decomposition of A are iso- morphic, or the only possible ones are2-groups and in this case O2(A)is not isomorphic to C₂C₂.

Clearly the supersolvability of Aut (A) is a necessary condition since Aut (A) is, up to isomorphism, a subgroup of Aut (G).

The paper is organized as follows. In the first two sections we analyze what happens ifAis a finitep-group, wherepis any prime number. In the former we focus our attention on the inner structure of Aut (G), while inthe latter onthe problem of supersolvability. Finally, inthe last sectionwe generalize the results to any finite nilpotent groupA.

1. The structure ofAut (G).

Let A be a finite p-group, C2 be the cyclic group of order two and G:ˆA wr C2the restricted wreath product ofAandC2. Then, by [3], ifA itself is not the group of order two, the base groupFofGis characteristic in G and, if we fix a complement C₂ˆ hti, we canexpress Aut (G) as a product

Aut (G)ˆKI;

where K is the subgroup of Aut (G) consisting of those automorphisms which fixtandIis the subgroup of Aut (G) consisting of those inner au- tomorphisms which correspond to conjugations by elements ofF. More- overIis normal in Aut (G), and sinceIis isomorphic to a quotient ofF,Iis a p-group. Let us give now another interpretation ofK. We identify the base subgroupFwithAA, the direct product of two copies ofA. Denote

as usual the elements ofFwith couples (x;y), wherex;y2A, and withd the involution of F defined by: d(x;y)ˆ(y;x) for each x;y2A. Let C_{Aut (F)}(d) be the centralizer ofdinAut (F).

LEMMA1. The groups K and CAut (F)(d)are isomorphic.

PROOF. Anisomorphism is givenby the map that sends h2K into p_Fhi_F2Kwherei_Fis the canonical injection ofFinGandp_Fthe canonical projectionofGonF. The inverse of this map is the application that sends W2Kinto the automorphismFofGso defined:F(a1a2t^e):ˆiG(W(a1;a2))t^e,

for eacha1a2t^e2G. p

We note that in this situation the automorphism d ofF is indeed the conjugation by the elementtthat generatesC₂inG.

To study more indetail the structure of the group K, we give some more notation that will be used through all the paper.

We indicate with:

Aut_Z(A);Aut_Z(F) the groups of the central automorphisms ofAand of F respectively (that is the groups of the automorphisms which act tri- vially onthe central factor groups A

Z(A)and F

Z(F)respectively).

K_Z:ˆK\Aut_Z(F) and similarly ifXis any subgroup of Aut (F) we use X_ZforX\Aut_Z(F).

D(Y):ˆ f(y;y)jy2Ygfor eachY subgroup ofA(we use simplyDfor

D(A)).

r(Y):ˆ h(y;y ¹)jy2Yifor each Y subgroup ofA(we use simplyr

forr(A)).

H:ˆC_K(D) the centralizer ofDinK.

L:ˆC_K(r) the centralizer ofrinK.

A:ˆ faf jf 2Aut (A)gwhereaf denotes the automorphism ofFdefined bya_f(a1;a2)ˆ(f(a1);f(a2)), for each (a1;a2)2F. (ClearlyA'Aut (A)).

Givenany elementWof Aut (F), define two endomorphismsW₁;W₂ofAby W(x;1):ˆ(W₁(x);W₂(x));8x2A:

As a Lemma we collect now some elementary facts. We omit the easy proofs.

LEMMA2. (i)IfW2K, then8x;y2A

W(1;x)ˆ(W₂(x);W₁(x)) and W(x;y)ˆ(W₁(x)W₂(y);W₂(x)W₁(y))

(ii) [ImW₁;ImW₂]ˆ1 and Aˆ(ImW₁)(ImW₂) (in particular ImW₁\

\ImW₂Z(A)andImW_iare normal subgroups of A).

(iii) Any element of K fixes the subgroups Dandr, in particular it permutes the generators(x;x ¹)ofr.

(iv) The subgroup H is the kernel of the group epimorphism F:W2K7 !W₁‡W₂2Aut (A)(whereW₁‡W₂is the map a7!W₁(a)W₂(a)for each a2A) and Kˆ[H]A is the semidirect product of H and A.

(v) H

HZis an elementary abelian2-group.

(vi) The map~ :W2H_Z7!~W2Aut_Z(A)defined byW(a)~ :ˆW₁(a²)a ¹is a homomorphism of groups.

At this point we distinguish the two casespevenandpodd.

Let us suppose first thatAis a finite 2-group.

Inthis situationthe map~defined in Lemma 2 (vi) is in general not in- jective. In fact its kernel is constituted by the elementsWofHZ such that W₁(a²)ˆa²for eacha2A, and so, sinceW₁‡W₂ˆid_A,W₂(a²)ˆ1 for each a2A. Then

Ker~ˆ fW2H_ZjW_jA2A²ˆid_jA²_A²g:

LEMMA3. The kernel of the map~is an elementary abelian2-group.

PROOF. LetWbe a non trivial element of the kernel of~andaanele- ment ofA. Then

W²(a;1)ˆ(W²₁(a)W²₂(a);1)

and so (W²)₁ˆW²₁‡W²₂ and (W²)₂ˆ0 the null endomorphism of A. Since W²2H, then(W²)₁‡(W²)₂ˆid_Aand so (W²)₁ˆid_AandWhas order 2. p

Lemmas 2, 3 and the fact that Aut (G)ˆKIimply the following THEOREM2. If A is a2-group, then the composition factors ofAut (G) are either composition factors ofAut (A)or cyclic groups of order2.

The same result canbe proved forp6ˆ2, infact ifpis odd, the map~of Lemma 2 (vi) is a monomorphism, thereforeHZis a normal subgroup ofH that canbe embedded inAut (A). So, according to Theorem 2, we can state

THEOREM3. If A is a p-group, with p any prime, then the composition factors of the group K are either composition factors ofAut (A), or cyclic groups of order2.

We next show that in the situationp6ˆ2 we canlift the subgroupH_Zup toH, and this will be of fundamental importance for our purposes.

Fix a Remak decompositionofA, sayAˆA1A2. . .An, whereAi

are indecomposable direct factors of A. For each subset I of the set f1;2;. . .;ng, we defin e X_I:ˆ hA_iji2Ii, Y_I:ˆ hA_jjj2=Ii, so that AˆX_IY_I, and denote withW_(XI;YI)orW_Ithe automorphism ofFdefined by

W_(XI;YI)(xy;x1y1):ˆW_I(xy;x1y1):ˆ(xy1;x1y) for eachx;x12XIy;y12YI.

It is immediate to verify thatW_I2H. We canthenconsider the subset of H Q:ˆ fW_IjI f1;2;. . .;ngg. It is not difficult to prove thatQ is an elementary abelian 2-subgroup ofHthat canbe generated by the elements W_fig,iˆ1;2;. . .;n(we leave the proofs to the reader).

Note that the subgroupQdepends on the Remak decomposition, sayA, of A chosen at first, so that it is convenient to write Q^A:ˆ

:ˆ fW^A_I jI f1;2;. . .;ngg instead of Q:ˆ fW_IjI f1;2;. . .;ngg. If we choose another direct decomposition ofA, sayB, thenwe obtainingeneral another elementary abelian 2-subgroup, sayQ^B. The relationbetweenQ^A andQ^B is explicitated inthe next

LEMMA4. Q^Aand Q^Bare conjugate in K by an element of A_Z. PROOF. Suppose the two different Remak decompositions of A are A:ˆ fAigⁿ₁ andB:ˆ fBigⁿ₁. By the Krull-Schmidt theorem there exists a central automorphismf ofAsuch that for eachiˆ1;2;. . .;n,f(Ai)ˆBi. Consider then the automorphisma_f ofF.a_f is a central automorphism ofF and for eachI f1;2;. . .;ng,

a_f¹W^B_Ia_f ˆW^A_I;

and soQ^AandQ^Bare conjugate by an element ofA_Z. p These arguments allow us to give the following description of the structure of the subgroupH.

LEMMA5. Chosen a Remak decompositionAof A, the subgroup H of K is the product HˆHZQ^A:

PROOF. By Lemma 2 we already know thatHZis normal inKand that H

HZis anelementary abelian2-group. Suppose thatWis anelement ofHof order a power of 2, say 2^m. If we callM:ˆKerW₂andN:ˆImW₂, we have thatAˆMN. Infact by Lemma 2,MandNare both normal subgroups ofA. In order to prove that their intersection is trivial, consider an element a2M\N, say aˆW₂(x). By aninductionargument we have that W^k(x;1)ˆ(W^k₁(x);W₂(x)^k), for eachk1. Inparticular forkˆ2^mwe obtain W₂(x)^2mˆ1 and since 2 does not dividejAj, we have that x2KerW₂, i.e.

aˆ1. SoAˆMNand this decomposition ofAallows us to consider the automorphismW_(M;N). A simple computationshows thatWW_(M;N) 2H_Z. In this way, according to the fact that the order of H

H_Zis a power of 2, we have that for anyWinHthere exists anelement ofHof the formW_(R;S)(for some decomposition(R;S) ofA) such thatWW_(R;S) 2H_Z.

Now we fix a Remak decompositionAofAand callQthe subgroupQ^A of H. If W is any element ofH, we proved that there exist s2HZ and W_(R;S)2Q^B, for some decompositionBofA, that is a refinement of (R;S), such that WˆsW_(R;S). Call f the central automorphism of A such that Qˆa_f¹Q^Ba_f, thenW^af ˆs^afW^a_(R;S)^f 2H_ZQ. But now we can write the ele- mentWasWˆ[W ¹;a_f¹]W^af, and, sinceHandK_Zare normal subgroups of K, [W ¹;a_f¹]2HZand soW2HZQ, i.e.HˆHZQ. p

2. Supersolvability ofK.

We now concentrate our study on the problem of the supersolvability of the groupK. We of course assume that Aut (A) is supersolvable.

As before, we consider first the casepˆ2.

With the considerations of section 2.1 it is not difficult now to establish whenAut (G) is supersolvable. Making use of [8], we know that the auto- morphism group of a 2-group, notC2C2, is supersolvable if and only if it is itself a 2-group. Therefore we have the following

THEOREM 4. Let A be a finite 2-group. If A is not isomorphic to C₂C₂, then Aut (G) is supersolvable if and only if Aut (A) is super- solvable. If A'C₂C₂, thenAut (G)is not supersolvable.

PROOF. For the caseAnot isomorphic toC2C2, Theorem 2 and the results in[8] tell us that Aut (G) is a 2-group and so it is supersolvable.

IfA'C2C2, Aut (G), being not a 2-group, is not supersolvable. (In this case the subgroup IAof Aut (G) is isomorphic to the symmetric

groupS4). p

The casepodd requires more work.

We recall that a groupTis said to bestrictly p-closed(wherepis any prime number) if T⁰T^{p 1} is a p-subgroup of T, or equivalently if Tˆ[O_p(T)]S with S an abelian group of exponent that divides p 1.

Strictly p-closed groups are supersolvable (see Baer [1]), and the auto- morphism group of a finitep-group (p6ˆ2) is supersolvable if and only if it is strictlyp-closed (see G. Corsi Tani [8]).

Before making any other hypothesis than the supersolvability of Aut (A), we prove a relevant result concerning the subgroupK_ZofK.

LEMMA 6. If A is a p-group (p6ˆ2), then, with the same notations as before:

(i) K_Zˆ[H_Z]A_Zˆ[L_Z]A_Z, semidirect products.

(ii) H_Z\L_Zˆ1.

(iii) H_Z'L_Z'a subgroup of A_Z normal in A.

(iv) IfAut (A)is supersolvable, then KZis supersolvable.

PROOF. (i)It is immediate by the fact that the applicationsF₁andF₂ from K_Z to Aut_Z(G) defined respectively by F₁(W):ˆW₁‡W₂ and F1(W):ˆW₁ W₂ (where W₁‡W₂(a)ˆW₁(a)W₂(a) an d W₁ W₂(a)ˆ

ˆW₁(a)(W₂(a)) ¹ for each a2A) are both epimorphisms of kernels re- spectivelyH_ZandL_Z.

(ii) Let W2H\L, then W(a²;1)ˆW(a;a)W(a;a ¹)ˆ(a;a)(a;a ¹)ˆ

ˆ(a²;1) and since 2 does not divide jAj, W is the identity map, and so H\Lˆ1.

(iii) We have already proved thatHZ canbe embedded inA. More- over, using a better argument, one can consider the subgroup T:ˆH_ZL_Z.Tis subgroup ofK_Z, which is normal in K, soT\A is a subgroup of A_Znormal inA. Using(i)and Dedekind's modular law, we obtainthatT\A'H_Z'L_Z.

(iv) By(i) we obtainthat K_Z HZ'K_Z

LZ'A_Z, and since H_Z\L_Zˆ1, if Aut(A) is supersolvable,K_Zis a supersolvable group. p Inparticular Lemma 6 applies whenA is abelianand Aut (A) super- solvable. Inthis case Aut_Z(F)ˆAut (F) an d soKˆK_Zis a supersolvable group.

Let us now concentrate on the caseA non-abelian, and study more in detail the structure of the subgroup KZ, with particular interest on the actionofK onK_Z.

LEMMA7. Let A be a non abelian finite p-group (p6ˆ2), withAut (A) supersolvable. Then:

(i) O_p⁰(K_Z)ˆ1 and K_Zˆ[O_p(K_Z)]S, where S is an abelian Hall p⁰ subgroup of K_Z.

(ii) KˆK_ZC_K(S).

(iii) [KZ;K]Op(KZ).

PROOF. (i)Since Aut (A) is supersolvable, another result of G. Corsi ([8]) implies that Op⁰(Aut (A))ˆ1, and so, as A_Z is normal in A and A'Aut (A),Op⁰(A_Z)ˆ1. Now the mapF1 defined in Lemma 6(i)is a homomorphism with kernelH_Z, and so we deduce thatO_p⁰(K_Z)ˆO_p⁰(H_Z).

By Lemma 6 again,H_Zis isomorphic to a normal subgroup ofA_Zand so O_p⁰(H_Z)ˆ1, i.e. O_p⁰(K_Z)ˆ1 and the Fitting subgroup Fit(K_Z) coincides with Op(KZ). The supersolvability of K and Schur-Zassenhaus theorem complete the proof of this step.

(ii)TakeS a Hall p⁰ subgroup ofK_Z. Using Frattini's argument we have thatKˆK_ZN_K(S). Inorder to prove thatN_K(S)ˆC_K(S), consider a decompositionofAasAˆXY, whereXandY6ˆ1 are respectively the product of the abelian and non-abelian direct factors ofA. Since Sis in particular a p⁰ subgroup of AutZ(F) we have, according to [5], FˆC_F(S)[F;S], where C_F(S) is the centralizer of S in F and [F;S]:ˆ hw ¹g(w)jg2S;w2Fiis abelian. Moreover, up to conjugation, we cansuppose that S contains the automorphisms aand b defined by a(xy;x~~y):ˆ(~xy;x~y),b(xy;x~~y):ˆ(x ¹y;x~¹~y), for each x;~x2X,y;~y2Y.

Consider now the subgroups D(X) an dr(X). One can easily prove that D(X)' r(X)'X; XXˆD(X) r(X); [F;a]ˆ r(X); and [F;b]ˆ

ˆXX:From this, we obtainthat [F;S]ˆXX. ThenN_K(S) fixes the subgroupsD(X);r(X) an dYY, and so any elementf of N_K(S) canbe seenas a triplef ˆ(f0;f1;f2), wheref0ˆfjYYis anautomorphism ofYY and f1ˆf_jD(X),f2ˆf_jr(X) are automorphisms of X. Inparticular the ele- ments ofSare of the formWˆ(id;W₁;W₂). Now takenanyf 2N_K(S) an d any W2Swe have that W^f ˆ(id;W^f₁¹;W^f₂²)2S andW₁;W^f₁¹;W₂ andW^f₂² canbe seenasp⁰ elements of Aut (X). Aut (X) is a supersolvable group, since it is, up to isomorphism, a subgroup of Aut (A), inparticular Aut (X) is strictlyp-closed. Therefore [W₁;f1] lies inOp(Aut (X)), butW₁lies ina Hall p⁰-subgroup of Aut (X) an df1normalizes this subgroup and so [W₁;f1] must be also ap⁰-element, therefore [W₁;f₁]ˆ1.

Similarly [W₂;f2]ˆ1, and sof centralizesS, an dK ˆKZCK(S).

(iii)Using the previous two steps we obtain

[K_Z;K]ˆ[Op(K_Z)S;Op(K_Z)C_K(S)]Op(K_Z): p As a consequence of this Lemma we obtain the following result on the subgroupH.

LEMMA8. The subgroup H of K is strictly p-closed.

PROOF. Using the previous Lemmas we obtain that H⁰ˆ[HZQ;HZQ]Op(HZ)Op(H):

So we just have to prove that H

Op(H)has exponent that dividesp 1.

Note that H

HZ is an elementary abelian 2-group and soO_p(H)ˆO_p(H_Z).

Using the facts thatHˆH_ZQ,Qis anelementary abelian2-group, and H

O_p(H)is abelian, we deduce that H O_p(H)

_{p 1}

ˆ HZ

O_p(H_Z)

_{p 1}

, and since H_Z9Aut (A),H_Zis strictlyp-closed and H

O_p(H)

_{p 1}

ˆ1.

Inparticular we proved that [H;H]Op(H). As [A;A]Op(A), we now just have to consider the action of A onH. At this point we make hypotheses onthe structure ofA. Suppose first of all thatA has no pairs of isomorphic direct factors in any of its Remak decompositions.Note that this hypothesis is consistent in the case A abelian, in fact a result of G. Corsi [8] shows that any abelian p-group (p6ˆ2) with supersolvable automorphism group necessary satisfies this condition.

A first consequence of this assumption is contained in the next

LEMMA9. Let A be a p-group (p6ˆ2), withAut (A)supersolvable and such that A has no isomorphic direct factors in its Remak decompositions, then

(i) N_K(Q)ˆC_K(Q).

(ii) KˆK_ZC_K(Q).

PROOF. (i)We letfbe anelement ofNK(Q) and prove thatfcentralizes Qby showing thatf centralizes the generatorsW_figofQforiˆ1;2;. . .;n.

Let us beginby observing thatfpermutes the elementsW_fig. Infact ifQis associated to the following Remak decomposition of A, AˆX₁

X2. . .Xn, the automorphismW_figinterchanges the elements ofXiXi

inAAand acts like the identity on the others. Similarlyf ¹W_figf changes the positions of the elements off ¹(X_i)f ¹(X_i) and acts trivially on the rest.

SinceX_iis indecomposable,f ¹(X_i) is to be such, an d sof ¹W_figfmust be an element ofQthat acts not trivially only on one pair of direct indecomposable factors, i.e.f ¹W_figf ˆW_fjgfor somejˆ1;2;. . .;n. Thereforefpermutes the elementsW_fig.

Inorder to prove thatiˆj, sin ceAhas no isomorphic direct factors, it is enough to show thatX_i'X_j. From the previous observationwe immediately deduce thatjX_ij ˆ jX_jj. Let us prove now thatf(X⁰_jX_j⁰)ˆX_i⁰X_i⁰.

We denote with [dF;W_fjg] the subgrouphd(u ¹)W_fjg(u)ju2Fi; then [dF;W_fjg]ˆ h(x;x ¹)jx2X_ji ˆ r(X_j)

and

X⁰_jX_j⁰ r(X_j)\F⁰(X_jX_j)\F⁰ˆX_j⁰X_j⁰

and so r(X_j)\F⁰ˆX_j⁰X_j⁰. Similarly r(X_i)\F⁰ˆX_i⁰X_i⁰. Now we claim thatf(r(X_j))ˆ r(X_i). Infact let (x;x ¹)2 r(X_j), thenby Lemma 2 (iii), f(x;x ¹)ˆ(y;y ¹) for somey2A. Let yˆuv with u2X_i,v2Y_i, then

(uv;u ¹v ¹)ˆf(x;x ¹)ˆfW_j(x;x ¹)ˆW_if(x;x ¹)ˆ(uv ¹;u ¹v) and so, since 2 does not divide the order ofA,f(x;x ¹)ˆ(u;u ¹). Since jXij ˆ jXjj, we have proved thatf(r(Xj))ˆ r(Xi), and from this we obtain that

X⁰_iX_i⁰ ˆ F⁰\ r(X_i)ˆf(F⁰)\f(r(X_j))ˆ

ˆ f(F⁰\ r(X_j))ˆf(X_j⁰X⁰_j) Now for the sake of simplicity we let

X:ˆD(X_i):ˆ f(x;x)jx2X_ig, Y:ˆD(Yi):ˆ f(y;y)jy2Yig, M:ˆf(D(X_j)) and

L:ˆf(D(Y_j)).

Since f fixes DˆD(A), MD\f(XjXj)ˆD(f(XjXj)) and by reasons of orders, MˆD(f(XjXj)). Similarly LˆD(f(YjYj)). More- over, AˆX_iY_iˆX_jY_j and this implies M'X_j;L'Y_j and DˆXY ˆML.

Let nowpXandpMbe the projections fromDrespectively onXand on M, and setF:ˆpMpX.Frestricted toM is anendomorphism ofM, so there existsn1 such thatFⁿ(M)ˆF^n‡1(M):ˆR. Now

M⁰ˆ(Df(XjXj))⁰ˆDf(XiXi)⁰ˆD(X_i⁰)ˆX⁰; and so

F(M⁰)ˆpMpX(X⁰)ˆpM(X⁰)ˆpM(M⁰)ˆM⁰: ThereforeRˆFⁿ(M)Fⁿ(M⁰)ˆM⁰, inparticularR/ M.

Now call SˆKerFⁿ, then S/ M, R\Sˆ1 an d MˆRS. Therefore MˆRS and S is abelianas RM⁰. Since M'Xj, M is in- decomposable, so eitherMˆRorMˆS, inboth cases we haveXj 'Xi. Infact ifMˆR, then MˆF(M), soM'p_X(M). But jMj ˆ jXj, an d so Xˆp_X(M)'M, i.e.X_i'X_j:Otherwise ifMˆSthen Mis cyclic, soX_j too. Using the fact thatf(X_j⁰X_j⁰)ˆX_i⁰X_i⁰, we have that

X⁰_iX_i⁰ˆf(X⁰_jX_j⁰)ˆ1

soX_iis abelian and since it is indecomposable, it is cyclic. By the factX_iand X_j are of the same order we deduce they are isomorphic.

(ii)As a consequence of Remak-Krull-Schmidt theorem, we have that for eachaf 2Athere exists aafc 2A_Zsuch thatQ^af ˆQ^afc, but this means that a_fa_fc¹2N_K(Q), and so A ˆN_A(Q)A_Z. Therefore KˆHA ˆ

ˆH_ZQN_A(Q)A_ZˆN_K(Q)K_Z, and by the previous LemmaKˆC_K(Q)K_Z. LEMMA10. [A;H]O_p(H):

PROOF. Using Lemmas 6 and 9, and the fact thatDZcentralizesHwe obtainthatK ˆA_ZC_K(Q):InparticularAˆA_ZC_A(Q):Then

[A;H]ˆ[A;H_ZQ][A;H_Z]^Q[A;Q][K;K_Z][A;Q]

[K;K_Z][A_ZH_A(Q);Q][K;K_Z][A_Z;Q][K;K_Z]O_p(K_Z);

and sinceHis normal inK, [A;H]O_p(H): p Now we are inconditionto prove a result aboutK.

THEOREM5. The group K is supersolvable if and only ifAut (A)is supersolvable and either A has no isomorphic direct factors (and in this case K is strictly p-closed), or AˆA₁A₂with A₁'A₂indecomposable.

PROOF. Suppose first that inany Remak decompositionofAthere are no pairs of isomorphic direct factors. Then from the previous Lemmas it is clear that Op(K)ˆOp(H)Op(A) an d K⁰Op(K). Since K

Op(K)ˆ

ˆO_p(H)A

Op(K) HO_p(A)

Op(K) , we have that the exponent of K

Op(K) divides the exponent of Aut (A)

Op(Aut (A))and so also (p 1):It follows thatK is strictlyp- closed and thus supersolvable.

Suppose now thatAˆA₁A₂ is the direct product of two isomorphic indecomposable groups. Since Aut (A) is supersolvable, this implies that theA_iare not abelian ([8]). In particularFhas no abelian direct factors too, and this implies thatKZis ap-subgroup ofK. Infact ifSis anyp⁰-subgroup ofKZ, using [5] we have thatFˆCF(S)[F;S], with [F;S] central, then Krull-Schmidt theorem implies [F;S]ˆ1, i.e. Sˆ1. So we have that K_ZO_p(K) an d [K_Z;K]O_p(K). Using the same notations as before, the subgroupQnow reduces tohW₁;W₂i, anelementary abeliangroup of order 4. We indicate with lany isomorphism from A1 to A2; linduces the fol- lowing involution in A L(g1g2):ˆl ¹(g2)l(g1) for each g12A1;g22A2. Moreover, L induces a_L2AK. We have that ja_Lj ˆ2 an d N_K(Q)ˆ ha_LiC_K(Q). From these we obtainthat [A;H]O_p(K)hdi, an d so, since [A;A]O_p(A)O_p(K) (as A is supersolvable with Op⁰(A)ˆ1) and [H;H][KZ;K]Op(K), we have thatK⁰Op(K)hdi Fit(K), the Fitting subgroup of K. Thenwe have that K

Fit(K)ˆ

ˆFit(H)A

Fit(K) HOp(A)

Fit(K) and so exp K Fit(K)

divides (p 1). This is en- ough to deduce the supersolvability ofK.

Finally we consider the case AˆA₁A₂B with A₁'A₂ in- decomposable groups andB6ˆ1. Inthis situationwe prove thatK is not supersolvable by showing that there is anelement of order two inthe Fitting subgroup ofKwhich does not commute with all the elements of orderp, an d this is a contradiction, sincepis the largest prime number that dividesjKj.

Using the same notation as in the previous step, let us consider the following subgroup ofK,R:ˆ hW₁aL;W₂i, (whereaL now acts like the identity auto- morphism onthe elements ofK). It is easy to see thatRis a dihedral group of order 8 and that its centre consists of the subgroup generated by dW_f1;2gˆ[W₁;a_L]. IfK were supersolvable then,O_p(K) would be a Sylowp- subgroup ofK, moreover K⁰Fit(K), and sodW_f1;2g2Fit(K) would com- mute with every element of orderp. Now letube a non trivial homomorphism

fromA₁

A⁰₁toZ(K) (remember that inthis situationAut (A) supersolvable and A1'A2imply thatA1is non-abelian).udefines the following automorphism ofA:U(g):ˆgu(g).Uhas orderpand it inducesa_U2K. Finally, a simple computationshows that [dW_f1;2g;a_U]6ˆ1 and this contradiction completes the

proof. p

3. Supersolvability ofAut (G).

This sectionis devoted to prove Theorem 1.

We first consider the case thatAis a finitep-group and then derive the result for finite nilpotent groups as a consequence.

THEOREM6. Let A be a finite p-group withAut (A)supersolvable.

If pˆ2,Aut (A wr C2)is supersolvable if and only if A6'C2C2. If p6ˆ2, Aut (A wr C₂) is supersolvable if and only if A has no iso- morphic direct factors in any of its Remak decomposition.

PROOF. Ifpˆ2 the result follows from Theorem 4.

Let nowp6ˆ2 and suppose first thatAhas no isomorphic direct factors.

ThenAut (A wr C2)ˆKI, by Theorem 5, is anextensionof a strictlyp- closed group by a normal p-group and so it is strictly p-closed and su- persolvable.

If otherwiseAhas isomorphic direct factors, inorder to have Aut (G) supersolvable, we must require that K is supersolvable, so Theorem 5 implies AˆA1A2 withA1'A2 indecomposable factors. Now the con- jugationg_a(whereais the generator ofC2) is not in the Fitting subgroup of Aut (A wr C₂), otherwise it will be a central element. Therefore if we con- sider the dihedral subgroup R:ˆ hW₁a_L;W₂i (with the same construction used inthe second step of Theorem 5), we have that C_R(I)ˆ1. Then R\Fit(Aut (A wr C₂))ˆ1, and so Aut (A wr C₂)

Fit(Aut (A wr C₂)) is not abelian and

Aut (A wr C₂) not supersolvable. p

PROOF OFTHEOREM1. The key idea of the proof consists in observing that Aut (G) is a central product of the groups Aut (A_pwr C₂), whereA_pis thep-component ofAandpvaries onthe set of prime divisorsp(A) ofjAj.

Use induction onjp(A)j.

Ifjp(A)j ˆ1, thenA is a prime power group and there is nothing to prove. Let jp(A)j 2 and write AˆPQ, where PˆOp(A) an d

QˆOp⁰(A) (p2p(A)). Using the same notations as in the previous sec- tions, we have again thatFis characteristic inGand

Aut (G)ˆIK:

MoreoverIˆIpIp⁰, where

I_p:ˆ fg_(p1;p2)2Inn(A wr C₂)j(p₁;p₂)2PPg

(Ip⁰ is defined in a similar way), and K:ˆCAut (F)(d). Since FˆAAis nilpotent Aut(F)ˆAut (PP)Aut (QQ) and we can write

KˆK_pK_p⁰

where Kp is the subgroup of Aut(W) of all the elements which fix each element ofQQand commute withdp(dp is the involution ofFthat in- terchanges the elements ofPPand acts like the identity on the rest). In a similar way is definedK_p⁰. Finally we have that

[Ip;Kp⁰]ˆ[Ip⁰;Kp]ˆ1:

So we obtainthat Aut (G) is a central product of the groups IpKp'Aut(P wr C2) an dIp⁰Kp⁰ 'Aut(Q wr C2). The result follows from

Theorem 6 and the inductive hypothesis. p

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Manoscritto pervenuto in redazione il 15 giugno 2005