factor groups

Please, if you are going to answer this question, include as much detail as you can so that I can follow what your doing. Thank you very much! Note: G' means derived (commutator) subgroup of G and Sn is symmetric group of degree n Please find G' in each case (a) G is abelian (b) G = Sn

isomorphism theory

Please, if you are going to answer this question, include as much detail as you can so that I can follow what your doing. Thank you very much! Note: S4 means symmetric group of degree 4 A4 means alternating group of degree 4 e is the identity Is there a group homomorphism $:S4 -> A4, with kernel $ = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}?

isomorphism theory

Please, if you are going to answer this question, include as much detail as you can so that I can follow what your doing. Thank you very much! Show that a group G is simple if and only if every nontrivial group homomorphism G -> G1 is one-to-one.

isomorphism theory

Please, if you are going to answer this question, include as much detail as you can so that I can follow what your doing. Thank you very much! Note: G =~ G1 means G is isomorphic to G1 If G/K =~ H, show that there exists an onto homomorphism $:G -> H with kernel $ = K

Rings

Please, if you are going to answer this question, include as much detail as you can so that I can follow what your doing. Thank you very much! Given r and s in a ring R, show that 1 + rs is a unit if and only if 1 + sr is a unit. (I think you can use the idea that s(1 + rs) = (1 + sr)s somehow)