Let G be a group and let H be a normal subgroup of G. Let m be the index of H in G (that is, the number of cosets of H). Prove that for any a we have am H.
(b) Give an example of group G, a subgroup H of index in, and an element a G such that am is not in H. (Of course, your subgroup H had better not be normal.)
(4) (a) Suppose G is a group of size 77 and H is a group of size 52. Prove that the only homomorphism from G to H is the trivial homomorphism.
(b) Give an example of a group G of size 77 and a group K of size 56 and a nontrivial homomorphism from G to IC.
Prove that if G is an abelian group and N is any normal subgroup of G then GIN is abelian.
(b) Prove that if G is any group and N is a subgroup of Z(G) such that GiN is cyclic then G must he abelian. [Note that since N is contained in the center of G, N is automatically normal in G and N is itself abelian.]
(c) Give an example of a group G and a normal subgroup N of G such that both N and G/N are abelian, hut G is not abelian.
(d) Give an example of a group G and a subgroup N of Z(G) such that GIN is abelian but G is not abelian.

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... alternating groups, permutation groups, group homomorphisms, conjugacy classes, normal subgroups, general linear ... is a normal subgroup of . ... is a homomorphism. ...

... x, of G satisfying the equation x²=e form a sub group H of ... G=Z, H=2Z, then H is a normal subgroup of Z ... But if ϕ is a homomorphism, then ϕ(gh)=(gh)^(-1)=h^(-1 ...

Proofs : Grorups, Subgroups and Indexes. ... before, G acts on the left cosets of H, and this action induces a homomorphism G --> S_n ... 3) Suppose H is a subgroup of G ...

... Normal subgroups, Second Theorem of Isomorphism, Conjugates and Cyclic Groups ... Therefore, N is a normal subgroup of G ... First, I claim that ϕ is a homomorphism. ...

... This shows how to complete a proof regarding a ... 1} Prove that SLn(k) is a normal subgroup of GLn ... Consider the homomorphism det : GLn (k ) → k . Notice that ker ...

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... phi is a group homomorphism. ker(phi) is the largest normal subgroup of G. For ... Abelian Groups, Non-Abelian Groups, Isomorphisms and Subgroups are investigated. ...