# Rank of a $p$-group

Here I've found two definitions of the rank of a $$p$$-group https://groupprops.subwiki.org/wiki/Rank_of_a_p-group. However, for the $$2$$-group $$\mathbb{Z}/\mathbb{Z}_{4}$$, the rank with the first definition would be $$2$$ but if I'm not wrong with the second it would be $$1$$. So my question is: am I wrong or is the webpage wrong?

I think that the second definition might be something like the maximum $$r$$ for which there exists an abelian subgroup such that the minimum size of a generating set whose elements have order $$p$$ is $$r$$. Would this be right?

• You are wrong. Why is it $2$ with the first definition? – the_fox Feb 20 at 22:50
• ah, I think I see my problem. I was forgeting that the groups had to be elementary. Thanks. – elescararriba Feb 20 at 22:55
• There are many notions non-equivalent of rank in this context. – YCor Feb 21 at 1:12

The rank of $$\mathbb Z/4\mathbb Z$$ is $$1$$ as the only elementary abelian subgroup is $$2\mathbb Z/4\mathbb Z$$ which has order $$2^1$$.