What is a group of type $(2,2)$?

Question

This is a statement in Serre's A Course in Arithmetic (p. 18).

If $p\ne2$, the group $\mathbb{Q}_p^*/\mathbb{Q}_p^{*2}$ is a group of type $(2,2)$.

What is a group of type $(2,2)$?

Answer 1

I expose the cyclotomic structure of ${\bf Q}_p$ in this previous answer (that is, I find the isomorphism class of ${\bf Q}_p^\times/({\bf Q}_p^\times)^n$ exactly for all primes $p$ and $n\ge1$). The simpler case ${\bf Q}_p^\times/({\bf Q}_p^\times)^2$ for odd $p$ is easier to deduce as being $C_2\times C_2$, the Klein four group. It would seem the "typing" system that Serre refers to has to do with either the $p$-primary or invariant factor decomposition of f.g. abelian groups (into cyclic groups whose orders are put into tuples), see Wikipedia for more details.