The paper reports the results of a search for pairs of groups of order that can be placed in the distance for the case when . The constructions that are used are of the general character and some of their properties are discussed as well.
We classify up to topological type nonorientable bordered Klein surfaces with maximal symmetry and soluble automorphism group provided its solubility degree does not exceed 4. Using this classification we show that a soluble group of automorphisms of a nonorientable Riemann surface of algebraic genus q ≥ 2 has at most 24(q-1) elements and that this bound is sharp for infinitely many values of q.
A subgroup of a finite group is said to be -supplemented in if there exists a subgroup of such that and is -permutable in . In this paper, we first give an example to show that the conjecture in A. A. Heliel’s paper (2014) has negative solutions. Next, we prove that a finite group is solvable if every subgroup of odd prime order of is -supplemented in , and that is solvable if and only if every Sylow subgroup of odd order of is -supplemented in . These results improve...
We show that any finite 2-group, whose abelianization has either 4-rank at most 2 or 8-rank 0 and whose commutator subgroup is generated by two elements, is metabelian. We also prove that the minimal order of any 2-group with nonabelian commutator subgroup of 2-rank 2 is .
Let G be some metabelian 2-group satisfying the condition G/G’ ≃ ℤ/2ℤ × ℤ/2ℤ × ℤ/2ℤ. In this paper, we construct all the subgroups of G of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we apply these results to study the capitulation problem for the 2-ideal classes of some fields k satisfying the condition , where is the second Hilbert 2-class field of k.
It is well known that a group G = AB which is the product of two supersoluble subgroups A and B is not supersoluble in general. Under suitable permutability conditions on A and B, we show that for any minimal normal subgroup N both AN and BN are supersoluble. We then exploit this to establish some sufficient conditions for G to be supersoluble.
In questo lavoro viene data una caratterizzazione di quei -gruppi nilpotenti di classe due ed esponente che sono speciali. Vengono inoltre studiate alcune costruzioni, automorfismi e sottogruppi abeliani di -gruppi speciali.
Let G be a finite group and p a prime. We consider an F-injector K of G, being F a Fitting class between Ep*p y Ep*Sp, and we study the structure and normality in G of the subgroups ZJ(K) and ZJ*(K), provided that G verifies certain conditions, extending some results of G. Glauberman (A characteristic subgroup of a p-stable group, Canad. J. Math.20(1968), 555-564).
We prove that if the average number of Sylow subgroups of a finite group is less than and not equal to , then is solvable or . In particular, if the average number of Sylow subgroups of a finite group is , then , where is the largest normal solvable subgroup of . This generalizes an earlier result by Moretó et al.