... Thus I need to see actual complete rigorous proofs so that I can make sure that I remember all of the necessary and ... Therefore, G is abelian. (b) Proof: If Z ...

... Thus we must have . Therefore, . Since , must be abelian and hence . 2. Proof: Since is a normal Sylow-p subgroup of , then is a unique Sylow-p subgroup of . ...

... 11 must be a cyclic group, thus we must have ker F = H . Since ker F is normal in G , then H is a normal subgroup of G . 7． Proof: G is an abelian group with ...

... 6. Proof: H is a "normal subgroup" of G if and only if for any g in G, g^(-1)Hg=H. For example, if G is an abelian group, then any subgroup of G is a normal ...

...Proofs: 1. If G is an Abelian group, then, for all a, b in G, (a^2)(b^2) = (ab)^2 (a^2)(b^2) = (aa)(bb) = a(ab)b [Associativity] = ab(ab) [Since G is Abelian...

... (b) Use the below result and part (a) to show that G must be isomorphic to Sym(S). Thus any non-Abelian group of order 6 is isomorphic to S_3. ...Proof: ...

... I need a detailed, rigorous proof of this with explanation ... This provides examples of proofs regarding commutative rings and ... Hence (RS , +) is an abelian group. ...