Subgroups Of Odd Index Research Articles

Two-arc-transitive graphs of odd order — II

It is shown that each subgroup of odd index in an alternating group of degree at least 10 has all insoluble composition factors to be alternating. A classification is then given of 2-arc-transitive graphs of odd order admitting an alternating group or a symmetric group. This is the second of a series of papers aiming towards a classification of 2-arc-transitive graphs of odd order.

Finite simple exceptional groups of Lie type in which all subgroups of odd index are pronormal

Abstract A subgroup H of a group G is said to be pronormal in G if H and H g {H^{g}} are conjugate in 〈 H , H g 〉 {\langle H,H^{g}\rangle} for every g ∈ G {g\in G} . In this paper, we determine the finite simple groups of type E 6 ⁢ ( q ) {E_{6}(q)} and E 6 2 ⁢ ( q ) {{}^{2}E_{6}(q)} in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.

Maximal solvable subgroups of odd index in symmetric groups

Maximal solvable subgroups of odd index in symmetric groups

On Pronormal Subgroups in Finite Simple Groups

A subgroup H of a group G is called pronormal if, for any element g of G, the subgroups H and Hg are conjugate in the subgroup they generate. Some problems in the theory of permutation groups and combinatorics have been solved in terms of pronormality, and the characterization of pronormal subgroups in finite groups is a problem of importance for applications of group theory. A task of special interest is the study of pronormal subgroups in finite simple groups and direct products of such groups. In 2012 E.P. Vdovin and D.O. Revin conjectured that the subgroups of odd index in all finite simple groups are pronormal. We disproved this conjecture in 2016. Accordingly, a natural task is to classify finite simple groups in which the subgroups of odd index are pronormal. This paper completes the description of finite simple groups whose Sylow 2-subgroups contain their centralizers in the group and the subgroups of odd index in which are pronormal.

Nonabelian Composition Factors of a Finite Group Whose Maximal Subgroups of Odd Indices Are Hall Subgroups

We obtain a description of nonabelian composition factors of a finite nonsolvable group in which any maximal subgroup of odd index is a Hall subgroup.

On the pronormality of subgroups of odd index in finite simple groups

A subgroup H of a group G is said to be pronormal in G if H and H g are conjugate in H, H g> for every g \in G . Some problems in finite group theory, combinatorics, and permutation group theory were solved in terms of pronormality. In 2012, E. Vdovin and the third author conjectured that the subgroups of odd index are pronormal in finite simple groups. In this paper we disprove their conjecture and discuss a recent progress in the classification of finite simple groups in which the subgroups of odd index are pronormal.

Classification of maximal subgroups of odd index in finite groups with alternating socle

A classification of maximal subgroups of odd index in finite groups with alternating socle is obtained.

Maximal subgroups of odd index in finite groups with simple linear, unitary, or symplectic socle

We give a classification of maximal subgroups of odd index in finite groups whose socle is isomorphic to one of the groups PSL n (q), PSU n (q), or PSp n (q) for n ≥ 13.

Characters of Maximal Subgroups ofM-Groups

In this paper, we will prove some results about the structure of M-groups. We want to study the degrees of primitive characters of subgroups of M-groups. Since the subgroups of M-groups need not themselves be M-groups, we know that the primitive characters of the subgroup need not be linear, and so, we want to study the relationship between the degrees of the primitive characters and the index of the subgroups. This line of study w x is motivated by Theorem B of 20 , where G. Navarro proved that if G is an M-group and H is a subgroup with p-index for some set of primes p , then every primitive character of H has p-degree. In this paper, we will be able to get a stronger relationship between the degrees of primitive characters and the indices of subgroups of M-groups. The result of Navarro led us to conjecture that if G is an M-group and H is any subgroup, then the degrees of all the primitive irreducible characters in H must divide the index of H in G. However, almost immediately after making this conjecture, we noticed that the group that w x Dade constructed in 1 is a counterexample to this conjecture. This example relies heavily on the properties of the prime 2, and so we decided to check the validity of the conjecture when we make an additional oddness assumption. When the group has a maximal subgroup of odd index, we have discovered the following result.

Etendre les representations autoduaies

Let k be an algebraically closed field of characteristic p. Let H be a normal subgroup of odd index, prime to p, in a finite group G. We prove that an indecomposable kH-module G-stable and selfdual can always be extended to G. If the kH-module is irreducible, only the odd index hypothesis is required.

A note on the thin finite simple groups

In a previous paper [ 11, the author described a new approach to nonabelian simple groups of characteristic 2 type in which every 2-local subgroup of odd index is solvable and has cyclic Sylow subgroups for all odd primes (in fact, we imposed these conditions only on certain special 2-local subgroups of odd index). .The approach was made possible by certain new general results concerning groups of characteristic 2 type, which divide into the following three categories: