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?

  • 2
    $\begingroup$ 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

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 $2^1$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.