How do you show that G is not an abelian?

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• Group Homomorphism and Abelian Groups

  Proving (=>) seems almost obvious since if it is abelian that means xyx^(-1)y^(-1) = xx^(-1)yy^(-1)=ee which is in the kernel. Please show how to do the reverse (<=) and show that phi is abelian.

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  Show, by considering the group G=S, or otherwise, that the corresponding result is false when G is not abelian.

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  57192 If is a group such that (ab)^2=a^2b^2 for all a,b belongs to G, show that G must be abelian. Or, Show that the group G is abelian iff (ab)^2=a^2b^2.

• Center of a Group and Abelian Groups

  In fact, Z (G) is always a normal subgroup of G (but you are not asked to prove this) Problem #4 (a) I show that Z(G)=G if and only if G is abelian. "=>": If Z(G)=G, then for any a,b in G, a and b are also in Z(G).

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  This shows how to prove that G is a group under matrix multiplication and also how to prove that a group is Abelian.

• Group Actions and Transitive Permutations

  Show that G acts(on the left) by conjugation on A by g(bar)a=gag^-1, where g is any representative of the coset g(bar). Give an explicit exxample to show that this action is not well defined if A is not abelian. Ps.

• Structure of Groups : Cauchy's Theorem, Order, Abelian Groups, Non-Abelian Groups, Isomorphisms and Subgroups

  (a) Show H is not normal in G. (Hint: If H is normal, then H is a subset of Z(G), and then it can be shown that G is Abelian) (b) Use the below result and part (a) to show that G must be isomorphic to Sym(S).

• Many problems on group theory

  If any of them are abelian: i. state the conditions under which a group is abelian ii. show that the group is abelian 6. there are only two groups of order four (Z 4and v). How many groups are there of the order five?