$\bigwedge^{2}T\cong\mathbb{Z}/2\mathbb{Z}$

Question

Hello I have the next issue

Consider $T$ the subgroup of diagonal matrices of $GL(2, 3) $ (group of invertible matrices of $2$ by $2$ over a field of three elements)

It is well known that $T$ is isomorphic to the Klein four group, therefore

$\mathbb{Z}/2\mathbb{Z} \cong H_{2}(T, \mathbb{Z})\cong\bigwedge^{2}T $

I would like to give isomorphisms from $\mathbb{Z}/2\mathbb{Z}$ to $\bigwedge^{2}T$

Any hint would be helpful, thank you!

Answer 1

Fix your well known iso $T \simeq \mathbb{Z}_2 \oplus \mathbb{Z}_2$. You get for free that this is a $\mathbb{Z}_2$-vector space isomorphism; hence you obtain a basis $\{\alpha, \beta\}$ for $T$.

Now map

$$ 1 \in \mathbb{Z}_2 \mapsto \alpha \wedge \beta. $$

This map is a $\mathbb{Z}_2$-linear monomorphism (the wedge on the RHS is a basis element of the exterior algebra) and by a dimension argument it must be an iso.