Product of finitely permutable groups

In this paper we show that if A1,A2,…,An are finitely many pairwise permutable abelian min-by-max subgroups of the group G such that G=A1…An, then J(G)=J(A1)…J(An), which J(G) finitely residual of a group G.

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Introduction

In 1940 G. Zappa (See [21] ) and in 1950 J. Szip (See [20] ) studied bout products of groups concerned finite groups. In 1961 O.H. Kegel (See [8] ) and in 1958 H. Wielandt (See [10] ) expressed the famous theorem, whose states the solubility of all finite products of two nilpotent groups. In 1955 N. Itô (See [7] ) found an impressive and very satisfying theorem for arbitrary factorized groups. He proved that every product of two abelian groups is metabelian. Besides that, there were only a few isolated papers dealing with infinite factorized groups. P.M. Cohn (1956) (See [18] ) and L. Redei (1950)(See [19] ) considered products of cyclic groups, and around 1965 O.H. Kegel (See [23,24] ) looked at linear and locally finite factorized groups. In 1968 N.F. Sesekin (See [16] ) proved that a product of two abelian subgroups with minimal condition satisfies also the minimal condition. He and Amberg independently obtained a similar result for the maximal condition around 1972 (See [17,1] ). Moreover, a little later the proved that a soluble product of two nilpotent subgroups with maximal condition likewise satisfies the maximal condition, and its Fitting subgroups inherits the factorization. Subsequently in his Habilitationsschrift (1973) he started a more systematic investigation of the following general question. Given a (soluble) product G of two subgroups A and B satisfying a certain finiteness condition x , when does G have the same finiteness condition x ?(See [17] )

For almost all finiteness conditions this question has meanwhile been solved. Roughly speaking, the answer is 'yes' for soluble (and even for soluble-by-finite) groups. This combines theorems of B. Amberg (See [1][2][3][4]6] ), N.S. Chernikov (See [5] ), S. Franciosi, F. de Giovanni (See [3, 6, 25-29] ), O.H. Kegel (See [8] ), J.C. Lennox (See [12] ), D.J.S. Robinson (See [9,14] ), J.E. Roseblade (See [13] ), Y.P. Sysak (See [30][31][32][33] ), J.S. Wilson (See [34] ), and D.I. Zaitsev (See [11,15] ). Now, in this paper, we study the residual finite group and min-by-max subgroups of the group G and its relations, and the end we prove that if ). ...J(A ) J(A J(G) n 1



For do this, in chapter 2 we express the elementary lemmas and Theorems and in chapter three we prove the main Theorem.

Preliminaries: (Elementary properties and Theorems.)

In this chapter we express the elementary Lemma and difinitons whose used in prove the Man Theorem in chapter 2.1. Theorem (See [11,12] ): If the soluble-by-finite group G=AB is the product of two polycyclic-by-finite subgroups A and B, then G is polycyclic-by-finite.

Proof: Assume that G it not polycyclic-by-finite. Then G contains an abelian normal section U/V which is either torsion-free or periodic and is not finitely generated. Clearly the factorizer of U/V in G/V is also a counterexample. Hence we may suppose that G has a triple factorization G=AB=AK=BK, Where K is an abelian normal subgroup of G which is either torsion-free or periodic. By Lemma 1.2.6(i) of [4] (See also [17] ) the group G satisfies the maximal condition on normal subgroups, so that it contains a normal subgroup M which is maximal with respect to the condition that G/M is not polycyclic-by-finite. Thus it can be assumed that every proper factor group of G is polycylic-by-finite. [15] ): Let the soluble group G=AB be the product of two subgroups A and B with finite abelian section rank. If at least one of the factors A and B has an ascending normal series with central or periodic factors, then G also has finite abelian section rank.

Theorem (See

Proof: See ( [4] , Theorem 4.6.10). [6] ): Let the group G=AB=AK=BK be the product of three nilpotent subgroups A, B, and K, where K is normal in G. If K is minimax, then G is nilpotent.

3. Theorem (See

Proof: See ( [4] , Theorem 6.3.4). [6] ): Let the group G=AB=AK=BK be the product of two subgroups A and B and a minimax normal subgroup K of G. (i) if A,B, and K are locally nilpotent, then G is locally nilpotent. (ii) If A, B, and K are hypercentral, then G is hypercentral.

4.Theorem (See

Proof: See ( [4] , Theorem 6.3.7).

Lemma:

Let the group G=AB be the product of two abelian subgroups A and B such that AG=BG=1. Then the following hold.

of G does not have nontrivial normal subgroups which are contained in A or B, so that in particular Z(X)=1.

(iv) The FC-centre of G is trivial.

Proof: (i) They Lemma 2.7 we have that

and so is also trivial. (ii)This follows from the first part of the proof of Lemma 2.9. (iii) Let N be a normal subgroup of X contained in A. Then G normalizes N, so that by (ii)

(iv) Let a be an element of F A  , where F is the FC-centre of G. Since G is abelian by Theorem 2.5, the mapping

6 Theorem: (See [22] ): Let the group G=AB 1  be the product of two abelian subgroups A and B, at least one of which has finite section rank. Then there exists a non-trivial normal subgroup of G contained in A or B.

Proof:

And its centre is trivial by Lemma 2.15(iii). The subgroups

are isomorphic, and hence both have finite section rank. By Theorem 2.12 the metabelian group X has finite abelian section rank, and hence is hypercentral by Theorem 2.14. In particular 1, Z(X)  a contradiction.

2.17 Theorem: (See [35] ): Let the group G=A1…At be the product of finitely many pairwise permutable abelian minimax subgroups A1,…,At. Then G is a soluble minimax group. Proof: Assume that the theorem is false, and let G= A1…At be a counterexample for which the sum

It follows that D G is a soluble minimax group. On the other hand, the factor group

Let Ji be the finite residual of Ai for every i=1,…,t. It follows from lemma 2.15 that j i J J is the finite residual of the soluble minimax group AiAj, so that it is abelian. Hence

then N/M has finite exponent and hence is finite. Therefore M is an infinite normal subgroup of



This contradiction proves the theorem.

Main Result:

In this chapter by used the Lemmas and Theorems of chaper 2, we prove the Basic theorem of this paper as follows.

Main Theorem:

Let the group G= A1…At be the product of finitely many pairwise permutable abelian min-by-max subgroups A1,…,At. Then G is a soluble min-by-max group and J(G)=J(A1)…J(At).

Proof: It follows from Theorem 2.17 that G is soluble minimax group, and hence J=J(G) is abelian. Put ) J(A J i i  for each i=1,…,t. Then This proves that G is a min-by-max group and J=L=J1...Jt.

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