Algorithmically finite groups

We call a group Galgorithmically finite if no algorithm can produce an infinite set of pairwise distinct elements of G. We construct examples of recursively presented infinite algorithmically finite groups and study their properties. For instance, we show that the Equality Problem is decidable in our groups only on strongly (exponentially) negligible sets of inputs.     

Approximability of finite rank soluble groups by certain classes of finite groups

For soluble groups of finite rank we obtain the necessary and sufficient condition to be a virtually residually finite p-group. We also prove that a soluble group G of finite rank is residually π-finite for some finite set π of primes if and only if it has no subgroups of type Q and the torsion radical of G is a finite group.     

Thef-length of finite groups

L. A. Shemetkov [1] defined the concept of the p-length of an arbitrary finite group, generalizing the concept of the p-length of a p-solvable finite group of Hall-Higman [2]. In this article we introduce a more general concept of thef -length of an arbitrary finite group with respect to a certain formationf, and bounds onf-lengths and p-lengths are given for arbitrary finite groups.Translated from Matematicheskii Zametki, Vol. 19, No. 5, pp. 745–754, May, 1976.     

Clones of finite groups

If G is a finite group whose Sylow subgroups are abelian, then the term operations of G are determined by the subgroups of G × G × G. Received March 5, 2004; accepted in final form November 11, 2004.     

Automorphisms of finite groups

To the memory of Anatolii Illarionovich Shirshov.     

On automorphism groups of some finite groups

We show that if n > 1 is odd and has no divisor p4 for any prime p, then there is no finite group G such that│Aut(G)│ = n.     

Solvability of finite groups

H is called an ? _p -embedded subgroup of G, if there exists a p-nilpotent subgroup B of G such that H _p ∈ Syl _p (B) and B is ? _p -supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use ? _p -embedded property of primary subgroups to investigate the solvability of finite groups. The main result is follows. Let E be a normal subgroup of G, and let P be a Sylow 5-subgroup of E. Suppose that 1 < d ? |P| and d divides |P|. If every subgroup H of P with |H| = d is ?₅-embedded in G, then every composition factor of E satisfies one of the following conditions: (1) I/C is cyclic of order 5, (2) I/C is 5′-group, (3) I/C ? A₅.