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**Groups which are factorized by subgroups of finite exponents.**
*(English)*
Zbl 1089.20012

Let the group \(G=AB\) be the product of a locally soluble subgroup \(A\) of finite exponent and a locally finite subgroup \(B\) such that for each prime \(p\), \(A\) or \(B\) has no elements of order \(p\). It is shown that if \(G\) is an \(RN\)-group (for instance if it is locally soluble or radical), then \(G\) is locally finite-soluble. If an \(RN\)-group \(G\) is factorized by finitely many pairwise permutable locally finite subgroups \(A_i\) with finite mutually coprime exponents \(t_i\), where \(i=1,\dots,n\), then \(G\) is locally finite with finite exponent \(t=t_1t_2\cdots t_n\).

It should be noted that S. V. Ivanov has announced that every countable group can be embedded in a group which is the product of two groups which have all their subgroups of prime order \(p\). This implies that an arbitrary group which is the product of two subgroups of finite exponent, may have infinite exponent.

It should be noted that S. V. Ivanov has announced that every countable group can be embedded in a group which is the product of two groups which have all their subgroups of prime order \(p\). This implies that an arbitrary group which is the product of two subgroups of finite exponent, may have infinite exponent.

Reviewer: Bernhard Amberg (Mainz)

### MSC:

20E15 | Chains and lattices of subgroups, subnormal subgroups |

20F50 | Periodic groups; locally finite groups |

20E22 | Extensions, wreath products, and other compositions of groups |

### Keywords:

groups of finite exponent; factorized groups; products of groups; permutable subgroups; factorizations; locally finite groups
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\textit{N. S. Chernikov}, Acta Appl. Math. 85, No. 1--3, 81--92 (2005; Zbl 1089.20012)

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### References:

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