Binomial Expansion in a Ring

I would like to prove the following: Let p be a prime. Show that in the ring Z-p (set of integers modulo p) we have (a+b)^p = a^p+b^p for all a, b in Z-p. The following hint was given: observe that the usual binomial expansion for (a+b)^n is valid in a commutative ring. Thanks for the help

Proofs in Group Theory

Five Problems: Let G=[FORMULA1], with operation given by multiplication modulo 14. 1) by computing the Cayley table of G, or otherwise, show the G is a group. You may assume that without proof multiplication modulo 14 is associative. 2) Prove that the subset H={1,9,11} is a subgroup of G 3) Compute the left cosets of G a ...continues

Herstein problem Sylow chapter

Find the possible number of 11-Sylow subgroups, 7-Sylow subgroups, and 5-Sylow subgroups in a group of order 5^(2)*7*11 ( 5 squared times 7 times 11) this is prob #21 pg 103 of Topics in Algebra

Hernsteins Sylow type

If G is a group of order 385 show that its 11-Sylow subgroup is normal and its 7-Sylow subgroup is in the center of G From prob #9 pg 102 Topics in Algebra

Hernsteins Sylow Prob in Topics...

If G is of order 108 show that G has a normal subgroup of order 3^k ( 3 to the k power), where k is greater than or equal to 2 On pg 102 #10 in Topics of Algebra

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 ques ...continues

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!