In this paper, we determine all of subgroups of symmetric group S4 by applying Lagrange theorem and Sylow theorem. First, we observe the multiplication table of S4, then we determine all possibilities of every subgroup of order n, with n is the factor of order S4. We found 30 subgroups of S4. The diagram of lattice subgroups of S4 is then presented. Mathematics Subject Classification: 20B30, 20...

Let G be an arbitrary finite group and fix a prime number p. The McKay conjecture asserts that G and the normalizer in G of a Sylow p-subgroup have equal numbers of irreducible characters with degrees not divisible by p. The Alperin-McKay conjecture is version of this as applied to individual Brauer p-blocks of G. We offer evidence that perhaps much stronger forms of both of these conjectures a...

Let G be a finite group and suppose that the set of conjugacy class sizes of G is f1;m;mng, where m; n > 1 are coprime. We prove that m 1⁄4 p for some prime p dividing n 1. We also show that G has an abelian normal p-complement and that if P is a Sylow p-subgroup of G, then jP 0j 1⁄4 p and jP=ZðGÞpj 1⁄4 p. We obtain other properties and determine completely the structure of G.

The Algebraicity Conjecture for simple groups of finite Morley rank, also known as the Cherlin-Zilber conjecture, states that simple groups of finite Morley rank are simple algebraic groups over algebraically closed fields. In the last 15 years, the main line of attack on this problem has been the so-called Borovik program of transferring methods from finite group theory. This program has led t...

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.

Let G be a finite group, p a fix prime and P a Sylow p-subgroup of G. In this short note we prove that if p is odd, G is p-nilpotent if and only if P controls fusion of cyclic groups of order p. For the case p = 2, we show that G is p-nilpotent if and only if P controls fusion of cyclic groups of order 2 and 4.

In 1955, Hall and Paige conjectured that any finite group with a noncyclic Sylow 2-subgroup admits complete mappings. For the groups GL(2, q), SL(2, q), PSL(2, q), and PGL(2, q) this conjecture has been proved except for SL(2, q), q ≡ 3 modulo 4. We prove the conjecture true for SL(2, q), q ≡ 3 modulo 4. © 2004 Elsevier Inc. All rights reserved.

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...