Normal Sylow Subgroups Research Articles

Orders of products of elements and nilpotency of terms in the lower central series and the derived series

In this paper we prove that if G is a finite group, then the k -th term of the lower central series is nilpotent if and only if for every \gamma_{k} -values x,y \in G with coprime orders, either \pi(o(x)o(y))\subseteq \pi(o(xy)) or o(x)o(y) \leq o(xy) . We obtain an analogous version for the derived series of finite solvable groups, but replacing \gamma_{k} -values by \delta_{k} -values. We will also discuss the existence of normal Sylow subgroups in the derived subgroup in terms of the order of the product of certain elements.

Normal Sylow subgroups and monomial Brauer characters

Normal Sylow subgroups and monomial Brauer characters

On the First Zassenhaus Conjecture and Direct Products

AbstractIn this paper we study the behavior of the first Zassenhaus conjecture (ZC1) under direct products, as well as the General Bovdi Problem (Gen-BP), which turns out to be a slightly weaker variant of (ZC1). Among other things, we prove that (Gen-BP) holds for Sylow tower groups, and so in particular for the class of supersolvable groups.(ZC1) is established for a direct product of Sylow-by-abelian groups provided the normal Sylow subgroups form together a Hall subgroup. We also show (ZC1) for certain direct products with one of the factors a Frobenius group.We extend the classical HeLP method to group rings with coefficients from any ring of algebraic integers. This is used to study (ZC1) for the direct product $G\times A$, where $A$ is a finite abelian group and $G$ has order at most 95. For most of these groups we show that (ZC1) is valid and for all of them that (Gen-BP) holds. Moreover, we also prove that (Gen-BP) holds for the direct product of a Frobenius group with any finite abelian group.

Finite groups whose set of numbers of subgroups of possible order has exactly 2 elements

Counting subgroups of finite groups is one of the most important topics in finite group theory. We classify the finite non-nilpotent groups G whose set of numbers of subgroups of possible orders n(G) has exactly two elements. We show that if G is a non-nilpotent group whose set of numbers of subgroups of possible orders has exactly 2 elements, then G has a normal Sylow subgroup of prime order and G is solvable. Moreover, as an application we give a detailed description of non-nilpotent groups with n(G) = {1, q + 1} for some prime q. In particular, G is supersolvable under this condition.

Fitting Height and Diameter of Character Degree Graphs

Let a graph Γ have bounded Fitting height (i.e., there is a bound on the Fitting heights of those groups whose character degree graph is Γ) and G be any solvable group with character degree graph Γ and Fitting height h(G). We improve Moretò's bound by proving that if no vertex in Γ is adjacent to every other one, then h(G) ≤4, else h(G) ≤6. As a consequence, if a solvable group G has character degree graph with diameter 3, then h(G) ≤4. Moreover, G has at most one non-abelian normal Sylow subgroup in this case.

Structure of a group with elements of order at most 4

We prove that every group in which the order of each element is at most 4 either possesses a nontrivial class 2 nilpotent normal Sylow subgroup or includes a normal elementary abelian 2-subgroup the quotient by which is isomorphic to the nonabelian group of order 6.

Construction of Finite Groups

We1999 Academic Pressintroduce three practical algorithms to construct certain finite groups up to isomorphism. The first one can be used to construct all soluble groups of a given order. This method can be restricted to compute the soluble groups with certain properties such as nilpotent, non-nilpotent or supersoluble groups. The second algorithm can be used to determine the groups of orderCopyright pn· qwith a normal Sylow subgroup for distinct primespandq. The third method is a general method to construct finite groups which we use to compute insoluble groups.

On groups and fuzzy subgroups

Abstract In this short communication, it is pointed out that the characterizations of abelian groups and normal Sylow subgroups in terms of fuzzy subgroups in Asaad and Abou-Zaid (1993) can be proved simply.

The λstructure of the green ring of GL(2, (Fp) in characteristic P, III

In this paper the reader is assumed to have taken notice of [I]. In [III] 1 we described the $lambda;, and s-, structure of the Green ring of GL(2,F p), and Sl(2,F p). We shall now construct a subring of the Green ring which is invariant for the $lambda;, and s-, operations. It is generated by all the indecomposables with odd-dimensional composition factors. This sheds another light on the results in the previous sections. We shall also study a certain quotient of the Green ring, which is in fact the Green ring of a certain subgroup of GL(2,F p) consisting of upper triangular matrices. The multiplication and the λ, s-, structure of this quotient Green ring is described. Moreover it is shown how this λ and s-, structure controls the deviation from being a λ, respectively s-, ring of the Green ring of any finite group with a normal Sylow subgroup of order p. The sequence of Adams operations for these groups is shown to be periodic, and the period reflects the internal p-structure of these groups.

Normal Sylow subgroups of linear groups

Normal Sylow subgroups of linear groups