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 and H ... *(Please see attachment for proper citation of formulas and for problems 4 and 5)

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