# Thread: Group homomorphisms and short exact sequences

1. ## Group homomorphisms and short exact sequences

I have no idea how to do this problem so I was hoping someone could show me how to do it? Or even at least how to do part (a). If anyone can help me out I would appreciate so much!

Let $N\underline{\triangleleft}G$ be a normal subgroup, and assume that there exists a group homomorphism $p:G\rightarrow N$ that is left inverse to the inclusion of $N$ in $G.$

(a) Show that there exists a group homomorphism
$$f:G\rightarrow N\times(G/N)$$
satisfying
$$(\forall n\in N)\ f(n)=(n, N)$$
and
$$(\forall g\in G)\ (pr_{2}\circ f)(g)=gN.$$
Here $pr_{2}$ denotes the projection to the second factor.

(b) Prove that the homomorphism $f$ you constructed in part (a) is in fact an isomorphism.

(c) Reformulate the questions and answers above in terms of short exact sequences.

2. ## Re: Group homomorphisms and short exact sequences

Define

$f(x)=(p( x),x N)$ for $x\in G$