A property of Generalized McLain groups

REND. SEM. MAT. UNIV. PADOVA, Vol. 121 (2009)

A Property of Generalized McLain Groups

AHMETARIKAN(*)

ABSTRACT- In this short note we show that ifSis a connected unbounded poset andR a ring withno zero divisors, then a generalized McLain groupG(R;S) is a pro- duct of two proper normal subgroups.

1. Introduction.

McLain groups were defined in [3] for the first time. These groups are characteristically simple and locally nilpotent with some further interest- ing properties.

LetSbe an unbounded partially ordered set (poset, in short) andRbe a ring with16ˆ0. Define the generalized McLain group G(R;S) as in [2].

Now every element ofG(R;S) can be uniquely expressed as 1‡Xⁿ

iˆ1

a_ie_aibi

whereai2R,ai;b_i2S,ai<b_iforiˆ1;. . .;randn2N.

In [5], some properties ofG(Fp;S) are considered for some orderings where F_pis the field of pelements. The generalized McLain groups are considered in a general context in [2] and the automorphism groups of these groups are considered in [1], [2] and [4].

[2, Theorem 7.1] gives a necessary and sufficient condition to beG(R;S) indecomposable. In this short note we ask the following question :

Does G(R;S) have proper normal subgroups K and N suchthat G(R;S)ˆKN?

Two elements a;b2S are called connected, if there are elements

a0;. . .;an2S suchthat a0ˆa;an ˆb and for each0i<n, eith er

(*) Indirizzo dell'A.: Gazi UÈniversitesi, Gazi EgÏitim FakuÈltesi, Matematik EgÏitimi Anabilim Dali.06500 Teknikokullar, Ankara, Turkey.

E-mail: arikan@gazi.edu.tr

aiai‡1orai‡1ai.Sis calledconnectedif every pair of elements inS is connected.

We shall prove the following:

THEOREM. LetSbe a connected unbounded poset and Ra ring with no zero divisors. Then M:ˆG(R;S) has proper normal subgroups K and N suchthat MˆKN, CM(K)6ˆ1 and CM(N)ˆ1. Furthermore if 1‡ce_jz2M, th en 1‡ce_jz2K or 1‡ce_jz2N.

2. Proof of the Theorem.

LEMMA2.1. Let S be a connected unbounded poset and R a ring with no zero divisors. Then every finite family of non-trivial normal subgroups of M intersects non-trivially.

PROOF. Obviously it is sufficient to prove the lemma for two proper non-trivial normal subgroups ofM. LetNandKbe suchsubgroups ofM.

Assume N\Kˆ1 and follow the proof of [2, Theorem 7.1] to reach a

contradiction. p

PROOF OF THE THEOREM. PutM:ˆG(R;S) and letwˆ1‡aeab with 06ˆa2R, K:ˆC_M(hw^Mi) and N:ˆ h(1‡ce_gd)^M:1‡ce_gd2= K;c2Ri.

Then we will prove that MˆKN. Since Z(M)ˆ1, we have K6ˆM.

Clearly hw^Mi ˆD

(1‡aeab)^1‡Pr

iˆ1aielia‡Ps

jˆ1bjebmj :a;ai;bj2R;1ir;1jsE

ˆ

1‡ae_ab X^r

iˆ1

aa_ie_lib‡X^s

jˆ1

ab_jeam_j‡X

1ir1js

aa_ib_je_limj :a;a_i;b_j2R;

1ir;1js

:

Let a<s<t<b, then we have 1‡de_st2K with06ˆa2R by [2, Lemma 2.2]. Since a generator 1‡cegdofNis not contained inK, it must be of the form 1‡cebdor 1‡cemd(m>b) or 1‡cegaor 1‡cegl(l<a) and its conjugates must be of the form:

(1‡cebd)^1‡Pr

iˆ1aielib‡Ps

jˆ1bjedmjˆ1‡cebd

X^r

iˆ1

caielid‡X^s

jˆ1

cbjebmj‡X

1ir1js

caibjelimj

162 Ahmet Ari.kan

or

(1‡cemd)^1‡Pr

iˆ1aielim‡Ps

jˆ1bjedmjˆ1‡cemd

X^r

iˆ1

aielid‡X^s

jˆ1

cbjemmj‡X

1ir1js

caibjelimj

or

(1‡cega)^1‡Pr

iˆ1aielig‡Ps

jˆ1bjeamjˆ1‡cega

X^r

iˆ1

caielia‡X^s

jˆ1

cbjegmj‡X

1ir1js

caibjelimj

or

(1‡cegl)^1‡Pr

iˆ1aielig‡Ps

jˆ1bjelmjˆ1‡cegl

X^r

iˆ1

caielil‡X^s

jˆ1

bjegmj‡X

1ir1js

caibjelimj:

For all termse_uethat appear in each case,u2= [a;b] ore2= [a;b]. Hence we have that there is no product of these elements which equals 1‡est, i.e., 1‡est2= N. HenceN6ˆMand obviouslyMˆKN.

Clearly we have CM(K)6ˆ1. Assume CM(N)6ˆ1, then CM(K)\

\C_M(N)6ˆ1 by Lemma 2.1. But sinceZ(M)ˆ1, this is a contradiction.

The final part of the theorem follows by the construction ofKandN. Now

the proof is complete. p

COROLLARY2.2. LetSbe a connected unbounded poset andRa ring with no zero divisors. PutM :ˆG(R; S)and letNbe the subgroup defined in the theorem. Then

CM(hx^Mi)=CM(hx^Mi)\N

is perfect for every generator x of M of the form1‡aeab(a2R).

COROLLARY2.3. LetSbe a connected unbounded poset andRa ring with no zero divisors. PutM :ˆG(R; S). ThenMhas a decomposable non- trivialepimorphic image.

The author is grateful to the referee for careful reading and many valuable suggestions.

A Property of Generalized McLain Groups 163

REFERENCES

[1] M. DROSTE- R. GOÈBEL,The automorphism groups of generalized McLain groups. Ordered groups and infinite permutation groups. Math. Appl. Kluver Acad. Publ. Dortrecht,354(1996), pp. 97-120.

[2] M. DROSTE - R. GOÈBEL, McLain goups over arbitrary ring and orderings.

Math. Proc. Cambridge Phil. Soc.,117, No. 3 (1995), pp. 439-467.

[3] D. H. MCLAIN, A characteristically simple group.Math. Proc. Cambridge Phil. Soc.,50(1954), pp. 641-642.

[4] J. E. ROSEBLADE, The automorphism group of McLain's characteristically simple group.Math. Zeit.,82(1963), pp. 267-282.

[5] J. WILSON, Groups with many characteristically simple subgroups. Math.

Proc. Cambridge Phil. Soc.,86(1979), pp. 193-197.

Manoscritto pervenuto in redazione il 13 novembre 2007.

164 Ahmet Ari.kan