# Thread: Group theory; is the following true?

1. ## 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?

2. ## 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,.)).

3. ## 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 gi that equals eG, the corresponding word in the hi must equal eH (this requirement actually includes 1) above).

in short the gi and the hi must satisfy the same RELATIONS.