Proof. Any element of odd order in a symmetric group is an even permutation, so the 3-Sylow and 5-Sylow subgroups of S5 lie in A5. Therefore it suffices to focus on A5. Since |A5| = 60 = 22 · 3 · 5, the 3-Sylow subgroups have size 3 and the 5-Sylows have size 5. Call the numbers n3 and n5. By Sylow III, n3 | 20 and n3 ≡ 1 mod 3, so n3 = 1, 4, or 10. The number of 3-cycles (abc) in A5 is 20, and...

Given a set r of permutations of an n-set, let G be the group of permutations generated by r. I f p is any prime, it is known that a Sylow p-subgroup P of G can be found in polynomial time. We show that the normalizer of P can also be found in polynomial time. In particular, given two Sylow p-subgroups of G, all elements conjugating one to the other can be found (as a coset of the normalizer of...

2 Topics in Group Theory 2 2.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Examples of Groups . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Elementary Properties of Groups . . . . . . . . . . . . . . . . 4 2.4 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 Co...

‎in this paper we give a new condition for a sylow $p$-subgroup of a finite group to control transfer‎. ‎then it is deduced a characteri-zation of supersoluble groups that can be seen as a generalization of the well known result concerning the supersolubility of finite groups with cyclic sylow subgroups‎. moreover a condition for a normal embedding of a strongly closed $p$-subgroup is given‎. ...

These notes are defining the notion of centric linking system for a locally finite group. If group G has countable Sylow p-subgroups, we prove that, with finiteness condition on set intersections p-completion its classifying space is homotopy equivalent to nerve system.

Let G be a finite group. We prove that if the number of Sylow 3-subgroups is at most 7 and 5-subgroups 1455, then solvable. This strong form recent conjecture Robati.

If P is a p-group for some prime p we shall write M (P ) to denote the set of all maximal subgroups of P and Md(P ) = {P1, ..., Pd} to denote any set of maximal subgroups of P such that ∩d i=1 Pi = Φ(P ) and d is as small as possible. In this paper, the structure of a finite group G under some assumptions on the c-normal or s-quasinormally embedded subgroups in Md(P ), for each prime p, and Syl...

We explicitly determine all of the transitive groups of degree p2, p a prime, whose Sylow p-subgroup is not isomorphic to the wreath product Z p Z p . Furthermore, we provide a general description of the transitive groups of degree p2 whose Sylow p-subgroup is isomorphic to Z p Z p , and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objec...