Showing a group is infinite and nonabelian given its presentation

[Cojate]

Homework Statement

The question is out of Hungerford's Algebra (Graduate Texts in Mathematics). Page 69,#7:

Show that the group defined by generators a,b and relations $a^2=e, b^3 = e$ is infinite and nonabelian.

Homework Equations

The Attempt at a Solution

My professor gave hints, suggesting that we construct an onto homomorphism using Van Dyck's Theorem. Here is a sketch of my proof:

Let $G = \langle a, b | a^2 = e, b^3 = e \rangle$. By Van Dyck's Theorem, there exists a unique onto homomorphism from G to $D_3$. Note that $D_3 = \langle a^i b^j : 0 \leq i \leq 1, 0 \leq j \leq 2 \rangle$. Thus G is nonabelian since $D_3$ is nonabelian.

To show that G is infinite consider $\alpha, \beta \in S_\mathbb{N}$, where α = (34)(67)... and β = (123)(456)... . Here o(α) = 2 and o(β) = 3, but $|\langle \alpha, \beta \rangle | = \infty$. Again, by Van Dyck's Theorem, there exists a unique onto homomorphism from G to $\langle\alpha, \beta \rangle$. Therefore G is infinite. $\blacksquare$



Why does the existence and uniqueness of an onto homomorphism to a group with these properties give the group G the desired properties?

 

[Adrmay]

I understand that Van Dyck's theorem and the fact that \langle\alpha, \beta \rangle is infinite, but why does this imply that G is nonabelian and infinite?