Herstein type Sylow prob

Let G be a group of n x n matrices over the integers modulo p, p a prime, which are invertable. Find a p-Sylow subgroup of G. topics of Algebra pg103 #20

Symmetries

Let S be a square, with vertices labelled (anticlockwise), 1,2,3,4. a symmetry of S is a rotation or reflection which preserves the square... a) List all the symmetries of the square. b) Let "a" be a rotation about the centre of the square... (See attachment for full question)

Homomorphisms

• Let G be a group and let a,b be two elements of G. The conjugate of b by a is, by definition, the element {see attachment}. The centralizer of a, denoted by {see attachment} the set of all elements g in G such that ga=ag. i) Find all possible conjugates f the permutation ... *Please see attachment for complete list of questions

Rings and Groups (Ring Theory, Quaternions, Homomorphisms, Matrices)

A number of questions involving rings and groups. Example: 3) Let R be a ring and [equationA]. Let [equationB] be the ring of n x n matrices with entries in R. What is the identity element of S? *(Please see attachment for complete list of problems)

Rings and subrings

Prove, using axioms for a ring, the following... see attached!