Ia-Automorphisms and Permutational Wreath Products
Let be an uncountable universal locally finite group. We study subgroups such that for every , .
In this paper some infinitely based varieties of groups are constructed and these results are transferred to the associative algebras (or Lie algebras) over an infinite field of an arbitrary positive characteristic.
In classifying certain infinite groups under minimal conditions it is needed to find non-simplicity criteria for the groups under consideration. We obtain some of such criteria as a consequence of the main result of the paper and the classification of finite simple groups.
In this paper we deal with the class of groups for which whenever we choose two infinite subsets , there exist two elements , such that . We prove that an infinite finitely generated soluble group in the class is in the class of -Engel groups. Furthermore, with , we show that if is infinite locally soluble or hyperabelian group then .
Fitting classes and injectors are discussed in the class of -groups. A necessary and sufficient condition for the existence of injectors is given; it is also shown that, when this condition holds, the injectors form a unique conjugacy class.