In this paper we consider finite loops and discuss the following problem: Which groups are (are not) isomorphic to inner mapping groups of loops? We recall some known results on this problem and as a new result we show that direct products of dihedral 2-groups and nontrivial cyclic groups of odd order are not isomorphic to inner mapping groups of finite loops.
We investigate the situation that the inner mapping group of a loop is of order which is a product of two small prime numbers and we show that then the loop is soluble.
If ℱ is a class of groups, then a minimal non-ℱ-group (a dual minimal non-ℱ-group resp.) is a group which is not in ℱ but any of its proper subgroups (factor groups resp.) is in ℱ. In many problems of classification of groups it is sometimes useful to know structure properties of classes of minimal non-ℱ-groups and dual minimal non-ℱ-groups. In fact, the literature on group theory contains many results directed to classify some of the most remarkable among the aforesaid classes. In particular, V....
We classify the finite soluble groups satisfying the following condition: if H is a subgroup of G and H is not nilpotent, then the Fitting subgroup of H is the centralizer in H of its derived subgroup H'.
New criteria of existence and conjugacy of Hall subgroups of finite groups are given.
In this paper we prove the following results. Let π be a set of prime numbers and G a finite π-soluble group. Consider U, V ≤ G and such that and . Suppose also is a Hall π-sub-group of some S-permutable subgroup of G. Then and . Therefore,the set of all S-permutably embedded subgroups of a soluble group G into which a given Hall system Σ reduces is a sublattice of the lattice of all Σ-permutable subgroups of G. Moreover any two subgroups of this sublattice of coprimeorders permute.
Multiplication groups of (finite) loops with commuting inner permutations are investigated. Special attention is paid to the normal closure of the abelian permutation group.