Isomorphism

Let T be defined on real two dimensional plain, and that: (x,y)T = (ax+by, cx+dy) ; a, b, c, d real constants. Prove that T is a vector space homomorphism. What value of a, b, c, d will T be an isomorphic or isomorphism?

Conjugacy Classes of A Finite Group Problem

G is a finite group with elements a and b. Let the conjugacy classes of these elements be A and B respectively and suppose |A|^2, |B|^2 < |G|. Prove that there is a non-identity element x in G s.t. x commutes with both a and b.

Conjugacy Classes of S_n Splitting in A_n

I really don't understand how this works *at all*! Suppose K is a conjugacy class of S_n (the symmetric group), with K consisting of even permutations. Suppose that x in K. Show that K splits into two conjugacy classes in A_n (the alternating group) iff x commutes with no odd permutation in S_n.

Isomorphism proof

If G ={a + b*sqrt2 | a,b rational} and H = {matrix a 2b, b a | a,b rational}, H is a 2 x 2 matrix - a 2b b a show that G and H are isomorphic under addition. Prove that G and H are closed under multiplication. I know I need to define the function map first, but I don't know what it is in this problem, let alone ...continues

proof problem, absract algebra

Prove that if G is a finite group of prime power order p^a, then the center Z(G) can not be the identity subgroup. I am having problems starting, middling, and ending this proof :)

sylow problem (Hernstein)

If G is of order p^(2)*q,p,q primes, prove that either a p-Sylow sub-group or a q-Sylow subgroup of G must be normal in G. this is prob 14 in sec 2.12 of Hernstein's Topics of Alg.

Abelian group problem

Let G be an Abelian group such that the operation on G is denoted additively. Show that {a is an element of G| 2a = 0} os a subgroup of G. Compute the subgroup for G =13.

Rings that are not Isomorphic

I need to prove that the rings 2Z and 3Z are not isomorphic. Using the method used here, I must also be able to show that the rings R (set of reals) and C (set of complex) are not isomorphic.

Nilpotent Elements of Commutative Rings

I need to prove the following: An element a of a ring R is nilpotent if a^n=0 for some n in Z+. Show that if a and b are nilpotent elements of a commutative ring, then a + b are also nilpotent.

Rings with Unity that forms a group

I need to prove the following: Show that if U is the collection of all units in a ring with unity, then is a group. A reminder was given to make sure to show that U is closed under multiplication. Thanks for the Help.