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Is there a name for the power of the exponent of a $p$-group? So, if $\mathrm{exp}(G):=\max\lbrace o(g)|g\in G\rbrace=p^k$ for some $k\in\mathbb{N}$, is there a name for the $k$? Additionally, is there a name for the power $n$ of the order of a $p$-group with $|G|=p^n$ for some $n\in\mathbb{N}$?

I am new to $p$-groups and I was also wondering, if it is even "interesting" to consider special cases of $n$ and $k$, for example, if both are prime as well. I am using the books by "Groups of Prime Power Order" by Yakov Berkovich et al. to get a feeling for the topic and they do not (so far) talk much about these terms $n$ and $k$.

My interest arises from an application in statistical mechanics which "enforces" certain powers.

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  • $\begingroup$ Would “exponent exponent” work? $\endgroup$
    – Emil Jeřábek
    Commented Sep 12, 2023 at 9:03
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    $\begingroup$ It's the $p$-logarithm of the the exponent, so maybe more "power in the exponent" than "power of the exponent". $\endgroup$
    – YCor
    Commented Sep 12, 2023 at 9:03
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    $\begingroup$ I don't think there's a name for $k$ and $n$, just for $p^k$ and $p^n$. But what I want to add is that if you want to learn about finite $p$-groups, may I suggest that a better place to start would be Chapter 5 of Gorenstein's "Finite Groups" - unless you're already beyond that point. I find the exposition there much better motivated. $\endgroup$
    – Dave Benson
    Commented Sep 12, 2023 at 9:04
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    $\begingroup$ It might be the "length"? in analogy with the length of a module over a ring. For an arbitrary finite group, it would count the number of Jordan-Hölder subfactors with multiplicity (thus the unique assignment that's 1 on simple groups and additive under extensions). $\endgroup$
    – YCor
    Commented Sep 12, 2023 at 13:06

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