quotient groups

• Sep 22nd 2009, 12:46 AM
jin_nzzang
quotient groups
question: Give an examples of a group G and a normal subgroup N satisfying the following properties; or if no example exists, state that. In either case, justify answers.

(a)G is abelian, and G/N is non-abelian
(b)G is cyclic, and G/N is not cyclic
(c)G/N is not isomorphic to any subgroup G.

could anyone help me with this problem??? please?
• Sep 22nd 2009, 12:59 AM
NonCommAlg
Quote:

Originally Posted by jin_nzzang
question: Give an examples of a group G and a normal subgroup N satisfying the following properties; or if no example exists, state that. In either case, justify answers.

(a)G is abelian, and G/N is non-abelian
(b)G is cyclic, and G/N is not cyclic
(c)G/N is not isomorphic to any subgroup G.

could anyone help me with this problem??? please?

no example exists for (a) and (b). for (c) an example is $G=\mathbb{Z}$ and $N=2\mathbb{Z}.$
• Sep 22nd 2009, 03:02 AM
jin_nzzang
Hi
thanks a lot for your help,, but could you please justify the answers so that i can understand it better? (at least one of them as an example ? )
• Sep 22nd 2009, 03:19 AM
Swlabr
Quote:

Originally Posted by jin_nzzang
Hi
thanks a lot for your help,, but could you please justify the answers so that i can understand it better? (at least one of them as an example ? )

Remember that your quotient group acts very much like your group - it has multiplication $(Nx)(Ny)=N(xy)$.

So, if the group is abelian then $(Nx)(Ny)=N(xy) = N(yx) = (Ny)(Nx)$.