Mathieu groups and its degree prime-power graphs

Let cd(G) be the set of irreducible complex character degrees of a finite group G. The degree graph of G related to cd(G) was defined. It was proved many finite simple groups (but not all Mathieu groups) are uniquely determined by their orders and degree graphs. We hope to define a new graph related to cd(G) such that more simple groups can be uniquely determined by their orders and this newly defined graphs. Here a degree prime-power graph is defined and it is proved that all Mathieu groups can be determined uniquely by their orders and degree prime-power graphs.     

The completion of solvable groups

It is proved that every noncyclic solvable free group with solvability length exceeding two has isomorphic completions.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 543–550, May, 1971.     

On a class of groups of prime-power order

LetG be a finitep-group,d(G)=dimH ¹ (G, Z _p) andr(G)=dimH ²(G, Z_p). Thend(G) is the minimal number of generators ofG, and we say thatG is a member of a classG _p of finitep-groups ifG has a presentation withd(G) generators andr(G) relations. We show that ifG is any finitep-group, thenG is the direct factor of a member ofG _p by a member ofG _p .     

The isomorphism problem for solvable groups

It is proved that the algorithmic isomorphism problem is solved negatively for solvable groups of length ≥7.     

The solvable radical of Sylow factorizable groups

Let G be a finite group. A complete Sylow product of G is a product of Sylow subgroups of G, one for each prime divisor of |G|. We shall call G a Sylow factorizable group if it is equal to at least one of its complete Sylow products. We prove that if G is a Sylow factorizable group then the intersection of all complete Sylow products of G is equal to the solvable radical of G. We generalize the concepts and the result to Sylow products which involve an arbitrary subset of the prime divisors of |G|. Received: 26 January 2005     

Universally equivalent solvable groups

Every group that is finitely presented in the varietyA ⁿ of solvable groups. and is universally equivalent to a free group F_r(A ⁿ) in this variety, is embedded in the Cartesian degree of F₂(A ⁿ). All subgroups on a set of two generators in that Cartesian degree which are universally equivalent to F₂(A ⁿ) are determined. Free solvable and nilpotent groups are proved universally equivalent. Supported by RFFR grant No. 99-01-00567, and through the RP “Universities of Russia. Fundamental Research.” Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 227–240, March–April, 2000.