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 prove that it is one-to-one and onto. Any help would be much appreciated.

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 :)