A remark about central automorphisms of groups

A Remark about Central Automorphisms of Groups.

GIOVANNICUTOLO(*)

To Guido Zappa on his 90th birthday

The aim of this paper is to prove and discuss the following simple result on central automorphisms, that, although elementary and not surprising at all, and therefore possibly known, seems to have not been recorded in the literature yet.

PROPOSITION. Let Z be a subgroup of the centre of a group G, and suppose that Z has finite exponent n. Also suppose that G has no non- trivial direct factor contained in Z. Let G be the group of those auto- morphisms of G acting trivially on G=Z. Then

(i) G has finite exponent dividing n;

(ii) Gacts nilpotently on G. More precisely, if k is the least positive integer such that n is not divisible by the (k‡1)th power of any prime, then[Z;_{2k 1}G]ˆ1;

(iii) Gis nilpotent of class at most2k 1, where k is defined as in(ii).

Of course the word `abelian' in the statement can be equivalently re- placed by `cyclic'. As it always happens when dealing with central auto- morphisms, some hypothesis to exclude the presence of certain kinds of abelian direct factors is necessary here, the reason being that every au- tomorphism of a direct factor of a group obviously extends to an auto- morphism of the group acting trivially modulo the factor itself. It is worth recalling that to every g2G there corresponds the homomorphism g :g2G7![g;g]2Z; if, as we are assuming, no nontrivial direct factor of Gis contained in Z, in the further hypothesis thatZ is finite Fitting

(*) Indirizzo dell'A.: UniversitaÁ degli Studi di Napoli ``Federico II'', Diparti- mento di Matematica e Applicazioni ``R. Caccioppoli'', Via Cintia - Monte S. Angelo I-80126 Napoli, Italy; e-mail address: [email protected]

URL: http://www.dma.unina.it/cutolo/

Lemma shows that the mappingg7!gdefines a bijection betweenG and Hom (G;Z) (see [1, 3]).

Part (i) of the Proposition can be compared to a theorem by Dixon and Evans [2], according to which ifGis a group andZ(G) has finite exponent thenGhas a central automorphism of infinite order if and only ifGhas an infinite abelian direct factor (in which caseGadmits an uncountable tor- sion-free abelian group of central automorphisms). Easy examples show that if Z(G) is only assumed to be periodic thenGmay have nonperiodic central automorphism even if it has no notrivial abelian direct factors. For instance, for any periodic abelian groupA of infinite exponent, one such example is the well-known class-2 nilpotent group Gˆ(AZ)^jhxi, where the assignment a7![a;x] gives an isomorphism fromA toZˆZ(G); if, instead,nˆexpAis supposed to be finite, then the same group provides an instance where the exponent ofG, as in part (i) of the Proposition, is exactlyn.

Next we shall prove the Proposition. Throughout we will assume thatG andZare chosen as in the hypothesis. The first remark is that ifpi s a pri me dividingnandZp⁰i s thep⁰-component ofZthenG=Zp⁰andZ=Zp⁰inherit the hypothesis from GandZ. In fact, i fA=Zp⁰ i s a di rect factor ofG=Zp⁰ con- tained inZ=Z_p⁰thenAˆZ_p⁰Bfor some subgroupB, and it follows thatB is a direct factor ofGcontained inZ, henceBˆ1 and soAˆZ_p⁰.

This remark allows us to reduce the proof of the Proposition to the case whenZi s ap-group.

The statement will be a consequence of the property established in the following Lemma, which is in fact equivalent to the absence of nontrivial direct factors ofGcontained inZ.

LEMMA1. In the given hypotheses, further suppose that Z is a p-group for some prime p. ThenV1 Z^pi

G⁰G^pi‡1for every natural number i.

PROOF. Let x2Z and suppose that y:ˆx^pi has orderp. If y2=N:ˆG⁰G^pi‡1 then hxi \Nˆ1 and xN has orderp^i‡1ˆexp (G=N).

Hence hxNiis a direct factor of G=N, and it follows thathxiis a direct factor ofG. This is excluded by the hypotheses, hencey2N.

From now on we will assume that Z is a nontrivial p-group and let nˆp^l.

To prove thatGⁿˆ1 we can argue by induction onl. Letmˆp^{l 1}. Then Z^mG⁰Gⁿ by the Lemma. Now G, as a group of central automorphisms, centralizesG⁰, and [Gⁿ;G]ˆ[G;G]ⁿˆ1, because [G;G]Z(G). Therefore

G centralizesZ^m, and so it acts trivially onZ=Z1, where Z1ˆVl 1(Z).

ThereforeG=CG(G=Z1) can be embedded in Hom (G=Z;Z=Z1), that has ex- ponentpat most, hence [G;G^p]Z1. By applying the induction hypothesis to G andZ₁ we see that the exponent ofG^p divides mˆexpZ₁, hence Gⁿˆ1. This proves part (i) of the Proposi ti on.

Now decompose Z as the direct product of homocyclic factors:

ZˆA1A2 At, where each of theAiis homocyclic of exponentp^ei and the integerse_iare such thatke1>e2> >et>0. For eachiwe have V₁(A_i)ˆA^p_i^{ei 1}G⁰G^pei, by the Lemma, hence [V₁(A_i);G] [G⁰G^pei;G]ˆ[G;G]^pei Z^pei, so that [V₁(A_i);G]V₁(A₁A₂ A_{i 1}). As a further consequence, for every positive integerje_i, si nce Vj(Ai)Z^{pei j},

[V_j(A_i);G]V_j(A₁A₂ A_{i 1})V_{j 1}(A_iA_i‡1 A_t)\Z^{pei j}

ˆV_j(A₁A₂ A_{i 1})V_{j 1}(A_i) Y^t

sˆi‡1

V_{j (ei es)}(A_s);

…†

whereVr(As) i s 1 i fr<0. For everyz2Zdefinez^tas follows: ifzˆ1 let z^tˆ2; if 16ˆz2A_ifor somei, letz^tˆi‡2n, wherepⁿ is the order ofz and finally, ifzˆQ^t

iˆ1ai, whereai2Aifor all i, letz^t ˆmaxfa^t_i j1itg.

We can also extend to scope of t to subgroups ofZ, by setting H^tˆmaxfh^tjh2Hg for allHZ. Now () shows that [H;G]^t<H^t for all nontrivial subgroupsHofZ(note thatz^t>2ˆ1^tfor allz2Zn1). Since Z^t ˆ2e₁‡12k‡1 it easily follows that [Z;_{2k 1}G]^t2, hence [Z;_{2k 1}G]ˆ1, as stated in (ii). It also follows that [G;_2kG]ˆ1, which

yields (iii). Thus the Proposition is proved. p

It would have also been possible to prove part (i) by using the equality [g;gⁿ]ˆQⁿ

iˆ1[g;_ig]… †ⁿⁱ , true for all g2G andg2G, where nˆp^lˆexpZ, and an argument like that employed for part (ii) to deal with the com- mutators of the form [g;_ig].

The bounds 2k 1 in parts (ii) and (iii) of the Proposition are actually attained, at least in the case when Zis not a 2-group. An example is the following. Letpbe a prime, letkbe an integer greater than 1 and let

Gˆ u₁;u₂;g₁;g₂;x

u^p_i ˆg^p_i^kˆx^pkˆ1; [ui;gi]ˆx^pk1 8i;j2 f1;2g [x;g_i]ˆ[x;u_i]ˆ[g₁;g₂]ˆ[u₁;u₂]ˆ[u_i;g_j]ˆ1; if i6ˆj

* +

:

ThenZ:ˆZ(G)ˆ hg^p₁;g^p₂;xihas exponentp^kandGhas no nontrivial abe- lian direct factor. Consider the central automorphismsaandbof Gdefined by:

a;b

u₁7!u₁

a:x7!g^p₁x u27!u2

g₁7!g₁g^p₂ b:x7!g^p₂x:

g27!g2x 8>

>>

><

>>

>>

:

It is easy to check that [x;_{2(k 1)}b]ˆx^{pk 1}6ˆ1; this shows that the bound i n (ii) is sharp. Ifpis odd it can be shown that [a;_{2(k 1)}b]6ˆ1, thus not only ha;bihas class 2k 1, but it is not even a 2(k 1)-Engel group. We sketch a proof of this fact. Every element ofG0:ˆ ha;biacts trivially onhu1;u2i, hence it is described by its action on the homocyclic grouphg₁;g₂;xi, and so by (the image modp^kof) a 33 matrix over thep-adic integers, taken with respect to the basis (g₁;g₂;x). Thusaandbare represented by

Aˆ

1 p 0

0 1 1 p 0 1 0

B@

1

CA and Bˆ

1 p 0

0 1 1

0 p 1

0

@

1 A:

Exactly as in the proof of the Proposition, decomposeZˆZ(G) as the direct product of the homocyclic groups A₁ˆ hxi (of exponentp^k) and A₂ˆ hg^p₁;g^p₂i(of exponentp^{k 1}), and define a weight maptaccordingly. It fol- lows that [hg1;xi;g_n(G0)]^t[hg1;xi;nG0]^t2k n‡1 and [hg2i;g_n(G0)]^t [hg2i;nG0]^t 2k n‡2 for all positive integersn<2k. Now, for every positive integer t<k, the elementsz2Z such that z^t2(k t) (resp.z^t2(k t)‡1) are those inV_{k t 1}(Z) (resp.V_{k t 1}(Z)hx^pti); hence, by settingnˆ2t‡1 above, we see that a matrix representing [a;_2tb] must have the form

Stˆ

tl1 tu12 tu13 tu21 tl2 tu23 tu_{31 t}u_{32 t}l₃ 0

B@

1 CA

where

tl_1tl_2tl₃1 …modp^t‡1†

tu₂₃0 …modp^t†

tu_ij0 …modp^t‡1†for all other values of iandj:

By direct (although not immediate) computation one can show that the (2;3)- and the (3;2)-entries of these matrices also satisfy the relations

tu32 ptu23(modp^t‡2); indeed, one sees that the corresponding con- gruence holds (for the appropriate value oft) for the matrix obtained by commuting twice either ofAorS_twithB. Once this has been established, it is not hard to check that _(t‡1)u₂₃4pd ¹_tu₂₃(modp^t‡2) for all integerst such that 1t<k, wheredˆ1 pis the determinant ofB. As1u23is not divisible by p² andp>2, it follows thatp^k does not divide_{(k 1)}u23, which amounts to saying that [a;_{2(k 1)}b] does not fixg₂, and so [a;_{2(k 1)}b]6ˆ1, as claimed.

REFERENCES

[1] J.E. ADNEY- T. YEN,Automorphisms of a p-group, Illinois J. Math.9(1965), pp. 137±143.

[2] M.R. DIXON- M.J. EVANS,On groups with a central automorphism of infinite order, Proc. Amer. Math. Soc.114(1992), no. 2, pp. 331±336.

[3] O. MUÈLLER,On p-automorphisms of finite p-groups, Arch. Math. (Basel)32 (1979), no. 6, pp. 533±538.

Manoscritto pervenuto in redazione il 22 dicembre 2005.