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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?

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

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