Construction of some simple locally finite groups Текст научной статьи по специальности «Математика»

УДК 517.2

Construction of Some Simple Locally Finite Groups

Ulviye B. Guven*

Department of Mathematics, Middle East Technical University, Ankara, 06 531

Turkey

Otto H. Kegel^

Mathematisches Institut, Eckerstr. 1, Albert-Ludwigs-Universitat Freiburg, Eckerstrabe, 1, Freiburg, D-79104, Germany

Mahmut Kuzucuoglu*

Department of Mathematics, Middle East Technical University, Ankara, 06 531

Turkey

Received 04.08.2013, received in revised form 15.09.2013, accepted 16.10.2013 We construct a proper class of simple locally finite groups. Namely for each infinite cardinal k, we construct uncountably many pairwise non-isomorphic simple locally finite groups of cardinality k, as a direct limit of finitary symmetric groups. The construction of the groups of similar kind for countably infinite order has been common knowledge as indicated in [2]. The countable ones are classified using the lattice of Steinitz numbers by Kroshko-Sushchansky in [3]. We give the classification of the uncountable ones by the pair, the cardinality of the group and the characteristic which corresponds to a Steinitz number. We study the structure of the centralizers of arbitrary elements in this new class of groups and correct some of the errors in the section about the centralizers of elements in S(£) in [3].

Keywords: simple locally finite group, Steinitz number, construction.

In [2] the construction of countably infinite order, simple locally finite groups as a direct limit of finite symmetric groups is given in the following way: Let n be the set of sequences consisting of prime numbers and £ G n. So £ = (pi,p2,...) is a sequence consisting of not necessarily distinct primes pi.

Let a G Sn. For a natural number p G N, a permutation dp(a) G Spn defined by (kn+i)d(a) = kn+ia, 0 ^ k ^ (p-1) and 1 ^ i ^ n is called a homogenous p-spreading of the permutation

c. So if a = ( 1""" n 1, then the homogeneous p-spreading of the permutation a is

dp(a) =

*1 ...ir 1 ...

*1 . ..

n

*n

n +1 n + *1

2n

n + in

(p — 1)n +1 (p — 1 )n + *1

pn

(p — 1 )n + in

We obtain direct systems by using homogenous pi-spreadings from the following embeddings where pi is the ith prime in the sequence £.

{1}

dp 1

and

+cbusra@metu.edu.tr ^otto.h.kegel@math.uni-freiburg.de ^ matmah@metu.edu.tr

Siberian Federal University. All rights reserved

{1} dPl A A A dA4

I-1-/ T =ni T JrxU2 ^-n3 T •••

where n = PiUi-i, i = 1, 2, 3... and Sni is the symmetric group on n* letters, Ani is the alternating group on n letters and no = 1. The direct limit groups obtained from the above direct systems are of strictly diagonal type and denoted by S(£) and A(£), respectively. Observe that S(£) < Sym(N).

In [2, Theorem 6.10] it is proved that the prime p £ £ repeats infinitely many times in £ if and only if the gro S(£) contains an isomorphic copy of the locally finite, divisible abelian group Cp~.

As the set of Steinitz numbers is uncountable, we have uncountably many pairwise nonisomorphic simple locally finite groups of the type S(£).

Recall that the formal product n = 2r2 3r3 5r5 ... of prime powers with 0 ^ rp ^ to for all primes p is called a Steinitz number (supernatural number). If a = 2r23r35r5 ... and p = 2s23s3 5s5 ... are two Steinitz numbers, then a|P if and only if rp ^ sp for all primes p. The set of Steinitz numbers forms a partially ordered set with respect to the above division. Moreover they form a lattice if we define the meet and the join as a A p = 2min{r2’s2}3min{r3’s3}5min{r5,s5} ... and a V P = 2max{r2>S2}3max{r3,S3}5max{r5 ,S5}

For each sequence £ we define Char(£) = plP1 p^P2 ... where rPi is the number of times that the prime p* repeat in £. If it repeats infinitely often, then we write p°°. Therefore for each £ £ n, there corresponds a Steinitz number Char(£). For the group S(£) obtained from the sequence £ we define Char(S(£)) = Char(£).

In [3] the class of groups S(£) is classified by using the lattice isomorphism between the lattice of Steinitz numbers ordered with respect to division of two Steinitz numbers and the lattice of

groups S(£) ordered with respect to being a subgroup.

By a type of a permutation a £ Sn we mean a vector t(a) = (kl, ..., kn) where k is the

number of cycles of length i in the cycle decomposition of a. By a principal beginning a0 of an element a £ S(£) we mean the element ao £ Sn where n is the smallest positive integer such that a0 cannot be obtained by a homogeneous spreading in the sequence £.

In [3] the structure of the centralizers of elements in S(£) is discussed and the following [3, Lemma 4.4] is stated. Let a £ S(£) with the principal beginning a0. If kl,... ,ks are non-zero components of t(a0), then

s

CS(0(a) =Dr (Cj S(£))

here I denotes the restricted wreath product of the cyclic group Ckj with S(£).

But it is well known that the center of the restricted wreath product is the trivial group

when the second group S(£) has infinite order see [1, Exercise 6.2.3 p.48] and [4, Corollary 4.4]. Therefore such a restricted wreath product cannot be a centralizer of a non-trivial element, as the element a must be in the center.

Moreover the characteristic of the group S(£) which appears on the right hand side as a component in the centralizer may change depending on the type of the element and the Char(£). The cyclic group Ckj should be Cj in [3, Lemma 4.4].

The corrected form of the centralizer of an element is the following.

Theorem 1. Let £ be an infinite sequence, g £ S(£) and type of principal beginning g0 £ Snk be t(g0) = (ri,r2,... ,rnk). Then

nk

CS(0(g) =DrCi(Ci? S(£i)) v ' i=l

Char(£)

where Char(£*) = -------------r* for i = 1,... nk. If r* = 0, then we assume that corresponding

Uk

factor is {1}.

For the construction of uncountable simple locally finite groups, let k be an arbitrary infinite cardinal number. Our methods are similar to [3]. Let FSym(«) denote the finitary symmetric group and Alt(«) denote the alternating group on the set k. As before, let n be the set of sequences of prime numbers and £ £ n. Then £ is a sequence of not necessarily distinct primes. Let a £ FSym(K), ( Alt(K) ). For a natural number p £ N, a permutation dp(a) £ FSym(Kp)

defined by (ks + i)dP(a) = ks + ia, i £ k and 0 ^ s ^ p — 1 is called homogeneous p-spreading of the permutation a. We divide the ordinal Kp into p equal parts and on each part we repeat

the permutation diagonally as in the finite case. So if a = ( 1... U ) £ FSym(K), then the

ii . . . in

homogeneous p—spreading of the permutation a is

dp(a) ' 1

ii

k(p — 1) + 1 ... k(p — 1) + n

k(p — 1) + ii ... k(p — 1) + in

K + 1 ... K + n

K + ii ... K + in

with the obvious meaning that the elements in Kp \ supp(dp(a)) are fixed.

We continue to take the embeddings using homogeneous p-spreadings with respect to the given sequence of primes in £. From the given sequence of embeddings, we have direct systems and hence direct limit groups FSym(K)(£) and ( Alt(K)(£) ) respectively. Observe that FSym(K)(£) and Alt(K)(£) are subgroups of Sym(Kw) where w is the first infinite ordinal.

Then we have the following Theorem.

Theorem 2. Let £ £ n be an infinite sequence.

(i) FSym(K)(£) = Alt(K)(£) if the prime 2 repeats infinitely often in £.

(ii) If the prime 2 repeats only finitely many times, then

|FSym(K)(£) : Alt(K)(£)| = 2

(iii) Alt(K)(£) is a simple locally finite group of cardinality k.

(iv) The permutations a, P £ FSym(K)(£) are conjugate in FSym(K)(£) if and only if the restrictions a! and P' of a and p are conjugate in FSym(Kni) for some i £ N.

The centralizers of elements in FSym(K) are known. Indeed if a £ FSym(K) with |supp(a)| = n, then a is contained in a subgroup of FSym(K) which is isomorphic to Sn. Assume that the type of a in this finite symmetric group is t(a) = (ri, r2,..., rn). Then

n

CfSym(n)(a) = (Dr (Ci I Sri)) x FSym(K \ supp(a)) i= 1

If ri = 0, then we assume that the corresponding factor in the direct product is {1}.

Now we state the structure of the centralizer of an arbitrary element in FSym(K)(£).

Theorem 3. Let £ be an infinite sequence. If a £ FSym(K)(£) with principal beginning a0 £ FSym(Kni), t(a0) = (ri,...,rn), and |supp(a0)| = n. Then

n

CFSym(n)(^)(a) = (Dr Ci(Ci ? S(£i))) x FSym(K)(£') i= 1

Char(£) Char(£)

where Char(£*) = -----r* and Char(£') = -. If r* = 0, then we assume that the

ni ni

corresponding factor in the direct product is {1}.

Lemma 1. The group CpTO is contained in FSym(K)(£) if and only if the prime p repeats infinitely often in £.

Proof. In the construction of the groups FSym(K)(£) when we take the direct limit of finitary symmetric groups, we also take the direct limit of finite symmetric groups as we did in the finite case Sn, ^ Sn . So for any infinite £, isomorphic copy of the groups S(£) are obtained as subgroups of FSym(K)(£). Therefore if the prime p repeats infinitely often in £, then by [2, Theorem 6.10] CpTO is a subgroup of FSym(K)(£).

On the other hand if the prime p repeats only finitely many times then one can imitate the proof of [2, Theorem 6.10] and show that CpTO is not a subgroup of FSym(K)(£). □

Hence by the above Lemma we may conclude that as the cardinality of the set of the Steinitz numbers is uncountable, the class of groups {AZt(«)(£) | £ G П} forms uncountably many

pairwise non-isomorphic simple, locally finite, non-linear groups of cardinality к for any given infinite cardinal к. Hence they form a proper class.

The groups FSym(K)(£) and AZ£(k)(£) are different from S(£) and A(£) by the cardinality properties whenever к is an uncountable cardinal. Observe also that FSym(K)(£) contains an isomorphic copy of S(£) as a subgroup. Since FSym(w)(£) and S(£) act on different sets clearly FSym(w)(£) is not a subgroup of S(£).

We may define Char(FSym(K)(£)) = Char(£) as before.

The following theorem gives the characterization of the groups FSym(K)(£) in terms of the lattice of Steinitz numbers and the cardinality к. Therefore for any given infinite cardinal к, there exist uncountably many pairwise non-isomorphic locally finite simple non-linear groups.

Theorem 4. Let к be a fixed infinite cardinal. There is a lattice isomorphism between the lattice of groups E = {FSym^)^) | £ G П } ordered with respect to being a subgroup and the lattice S of Steinitz numbers ordered with respect to division in Steinitz numbers.

Proof. One may prove this along the lines of [3, Theorem 2]. □

Acknowledgements: The third author would like to express his sincere thanks to the colleagues and the staff of FRIAS for warm hospitality during his visit to Freiburg.

References

[1] M.I.Kargapolov, Ju.I.Merzljakov, Fundamentals of the theory of groups, Graduate Text in Mathematics, 62, Springer-Verlag, 1979.

[2] O.H.Kegel, B.A.F.Wehrfritz, Locally Finite Groups, North-Holland Publishing Company, Amsterdam, 1973.

[3] N.V.Kroshko, V.I.Sushchansky, Direct Limits of symmetric and alternating groups with strictly diagonal embeddings, Arch. Math., 71(1998), 173-182.

[4] J.D.P.Meldrum, Wreath products of groups and semigroups, Pitman Monographs and Surveys in Pure and Applied Mathematics, 74, Longman, Harlow, 1995.

Построение некоторых простых локально-конечных групп

Юлвие Б. Гювен Отто Н. Кегель Махмуд Кузучуоглу

В статье строится собственный класс простых локально конечных групп, а именно, для каждого бесконечного кардинального числа к мы строим несчетное множество попарно неизоморфных простых локално конечных групп мощности к, как индуктивный предел финитарных симметрических групп. Как указано в [2], кострукция групп такого типа бесконечной счетной мощности хорошо известна. В счетном случае они классифицированы Крошко-Сущанским [3] с помощью решетки чисел Стейница. Мы классифицируем их в несчетном случае по мощности группы и характеристике, соответствующей числу Стейница. Мы исследуем структуру централизаторов произвольных элементов в группах из этого нового класса и исправляем некоторые ошибки в параграфе о централизаторах элементов из S (£) в [3].

Ключевые слова: простые локально-конечные группы, числа Стейница, построение.