Group theory; is the following true?

Hi guys,

I'm wondering if the following, not a homework assignment, is true:

Let G be a group, generated by three elements, $\displaystyle G= \langle g_1, g_2 , g_3 \rangle$ and let H be a group such that $\displaystyle |H|=|G|$ and $\displaystyle H=\langle h_1,h_2,h_3 \rangle$. The map $\displaystyle \phi : G\to H$ which satisfies $\displaystyle \phi(g_i)=h_i$ for i=1,2,3 is an isomorphism.

It is a sort of lemma I wish to be true in order to prove something else.

Can anyone help?

Re: Group theory; is the following true?

Hey Lockdown.

The injective nature is there, but you have to show that a homomorphism exists. Can you do some re-arranging to show this? (I.e. setup phi(g1*g2) = phi(g1) . phi(g2) for (G,*) and (H,.)).

Re: Group theory; is the following true?

it is not, in general, true. you need two more additional requirements:

1) $\displaystyle |h_i| = |g_i|$ for each i

2) if x is a word in the g_{i} that equals e_{G}, the corresponding word in the h_{i} must equal e_{H} (this requirement actually includes 1) above).

in short the g_{i} and the h_{i} must satisfy the same RELATIONS.