# Homomorphism 4

• Apr 24th 2009, 06:50 AM
mpryal
Homomorphism 4
If G1, G2 are two groups and G = G1 x G2 = {(a,b)|a in G1, b in G2, where we define (a,b)(c,d)=(ac,bd), show that:
a) N={a,e2)|a in G1}, where e2 is the unit element of G2, is a normal subgroup of G.
b) there is an isomorphism from N onto G1.
c) there is an isomorphism from G/N onto G2.

Please show steps. Thank you!
• Apr 24th 2009, 09:28 AM
ThePerfectHacker
Quote:

Originally Posted by mpryal
If G1, G2 are two groups and G = G1 x G2 = {(a,b)|a in G1, b in G2, where we define (a,b)(c,d)=(ac,bd), show that:
a) N={a,e2)|a in G1}, where e2 is the unit element of G2, is a normal subgroup of G.
b) there is an isomorphism from N onto G1.
c) there is an isomorphism from G/N onto G2.

Please show steps. Thank you!

For (a) show $\displaystyle gNg^{-1} = N$ for all $\displaystyle g\in G$.
For (b) define $\displaystyle f: N\to G_1$ by $\displaystyle f(a,e_2) = a$.
For (c) define $\displaystyle f: G\to G_2$ by $\displaystyle f(a,b) = b$ now use fundamental homomorphism theorem.