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.

Please see the attached file for the complete solution.
Thanks for using BrainMass.

Problem #3
(a) Since is a normal subgroup, then is a quotient group and we have natural homomorphism , for any . Since , then . Thus . This implies that . Therefore, for any , we must have .
(b) If is not ...

Solution Summary

Homomorphisms are investigated. The solution is detailed and well presented.

Subgroup proof. Let A and B be finite subgroups of G. Even though AB need not be a subgroup of G, show ... Now, define a map (not necessarily a homomorphism) . ...

... This provides examples of several proofs regarding subgroups in PDF ... H . This action gives rise to a homomorphism ϕ : G ... G/K is isomorphic to a subgroup of SX ...

... 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 ...

... 2 If H and K are normal subgroups of a ... In other words, has a subgroup isomorphic to , which means ... try and use the isomorphism theorem, where is a homomorphism. ...

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 ...

... Show that SL(n, R) forms a normal subgroup of GL ... n, R) 2) Define a group of homomorphism fronm GL ... matrices that commute with all over invertible matrices) Proof: ...

... Sylow P-Subgroups, Isomorphisms and Solvable Groups are investigated ... gM=M. So ker(f)={M}. By the first homomorphism theorem, G ... to f(G/M), which is a subgroup of ...

... Problem #2 Proof: Suppose is the internal direct product of two subgroups... Suppose is the internal direct product of two subgroups and ... First, is a homomorphism. ...

... of G2 , and that the kernel of h is a normal subgroup of G1. ... All I can say is to look at how I construct these proofs and to ... Definition -- Homomorphism: A map. ...

... 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 ...