We give a classification of irreducible characters of finite groups of Lie type of p′-degree, where p is any prime different from the defining characteristic, in terms of local data. More precisely, we give a classification in terms of data related to the normalizer of a suitable Levi subgroup, which in many cases coincides with the normalizer of a Sylow p-subgroup. The McKay conjecture asserts...

A regular map M is a cellular decomposition of a surface such that its automorphism group Aut(M) acts transitively on the flags of M. It can be shown that if a Sylow subgroup P ≤ Aut(M) has order coprime to the Euler characteristic of the supporting surface, then P is cyclic or dihedral. This observation motivates the topic of the current paper, where we study regular maps whose automorphism gr...

Let H be a subgroup of a finite group G. We say that H is SS-semipermutable in Gif H has a supplement K in G such that H permutes with every Sylow subgroup X of Kwith (jXj; jHj) = 1. In this paper, the Structure of SS-semipermutable subgroups, and finitegroups in which SS-semipermutability is a transitive relation are described. It is shown thata finite solvable group G is a PST-group if and on...

Valuations on a field K are encoded in the absolute Galois group GK of K: They are in one-to-one correspondence to the conjugacy classes of decomposition subgroups of GK which (apart from few exceptions) can be characterized in group theoretic terms. Roughly speaking decomposition subgroups of GK are maximal subgroups of GK with a Sylow-subgroup containing a non-trivial abelian normal subgroup....

Sylow subgroups are fundamental in the design of asymptotically efficient group-theoretic algorithms, just as they have been in the study of the structure of Ž . finite groups. We present efficient parallel NC algorithms for finding and conjugating Sylow subgroups of permutation groups, as well as for related problems. Polynomial-time solutions to these problems were obtained more than a dozen ...

A subgroup H of a group G is called ss-quasinormal in G if there is a subgroup B of G such that G = HB and H permutes with every Sylow subgroup of B; H is called weakly s-permutable in G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ≤ HsG, where HsG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We fix in every non-cyclic Sylow subgr...

A complete mapping of a group G is a permutation φ : G → G such that g 7→ gφ(g) is also a permutation. Hall and Paige proved that a finite group admits a complete mapping only if its Sylow-2 subgroup is trivial or non-cyclic. They conjectured that this condition is also sufficient. We prove that it is sufficient to check the conjecture for the 27 sporadic simple groups.

There are a number of results of group theory presented in Herstein [1] (the counting principles in Sections 2.5 and 2.11, Cayley’s theorem in Section 2.9 and the Sylow theorems of Section 2.12, among others) whose proofs follow similar ideas. However, it is not apparent that this is so. This handout covers the ideas necessary to tie these results together. The important idea is that of an acti...

Given an LCD group code C in a group algebra KG, we inspect kinship between C and G, more precisely between the subgroup structures of G and C. When C is MDS, the inter relation between K and G becomes more impressive. Application of Sylow theorem facilitated us to explore the inter relation between G and K (when C is LCD and MDS) in more general way and finally we get the result of Cruz and Wi...

In 1963 Ryser conjectured that there are no circulant Hadamard matrices of order > 4 and no cyclic difference sets whose order is not coprime to the group order. These conjectures are special cases of Lander’s conjecture which asserts that there is no abelian group with a cyclic Sylow p-subgroup containing a difference set of order divisible by p. We verify Lander’s conjecture for all differenc...