Abelian Sylow subgroups in a finite group

A New Finite Simple Group with Abelian 2-sylow Subgroups.

A 2-Sylow subgroup of J is elementary abelian of order 8 and J has no subgroup of index 2. If r is an involution in J, then C(r) = (r) X K, where K _ A5. Let G be a finite group with the following properties: (a) S2-subgroups of G are abelian; (b) G has no subgroup of index 2; and (c) G contains an involution t such that 0(t) = (t) X F, where F A5. Then G is a (new) simple group isomorphic to J...

The Sylow Subgroups of the Symmetric Group

In the Sylow theorems f we learn that if the order of a group 2Í is divisible hj pa (p a prime integer) and not by jo*+1, then 31 contains one and only one set of conjugate subgroups of order pa, and any subgroup of 21 whose order is a power of p is a subgroup of some member of this set of conjugate subgroups of 2Í. These conjugate subgroups may be called the Sylow subgroups of 21. It will be o...

The Total Number of Subgroups of a Finite Abelian Group

In this note, steps in order to write a formula that gives the total number of subgroups of a finite abelian group are made.

Cohomology of groups, abelian Sylow subgroups and splendid equivalences

Let G be a finite group and let R be a complete discrete valuation domain of characteristic 0 with residue field k of characteristic p and let S be R or k. The cohomology rings H∗(K,S) for subgroups K of G together with restriction to subgroups of G, transfer from subgroups of G and conjugation by elements of G gives H∗(−, S) the structure of a Mackey functor. Moreover, the group HSplenS(K) of ...

Finite abelian subgroups of the Cremona group of the plane