Monoidal群的LFCF－根

The LFCF-Radical of Monoidal Groups

群Ｇ称为Ｍｏｎｏｉｄａｌ群，如果任一非空集，总有１Ｇ∈Ｓ，群Ｇ称为ＬＦＣＦ－群，如果Ｇ有正规子群列，使得Ｆ为局部有限群，Ｃ／Ｆ为循环群，Ｇ／Ｃ为有限群．本文证明在Ｍｏｎｏｉｄａｌ群中，ＬＦＣＦ一性是根性．并且若Ｒ是Ｍｏｎｏｉｄａｌ群Ｇ的ＬＦＣＦ－根，则Ｒ与Ｇ／Ｒ中至少有一个是周期群．在模Ｇ的最大正规局部有限子群的情况下，Ｇ的可解性，局部可解性与ＬＦＣＦ－性等同．

A group G is called monoidal if for its every non-empty subset S, S2=S always implies 1G∈S. AnLFCF-group G is a group which has a normal series such that F is locally finite,C/F is cyclic,and G/C is finite. It is obtained in this paper that the LFCF-property is radical in the class of monoidal groups.And let R be the LFCF-radical of a monoidal.greup G, then either R Or G/R is torsion. Moreover, factoring outby the maximal normal locally finite subgroup of G, the solubility, locally solubility and LFCF-property are equivalent.