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Let $G$ be a finite $p$-group of nilpotency class $c$ and of derived length $d$. As is well known, we have $d\leq \lfloor\log_2 c\rfloor+1$,

(https://groupprops.subwiki.org/wiki/Derived_length_is_logarithmically_bounded_by_nilpotency_class).

Does there exist any information or any classification of finite $p$-group $G$, where $d=\lfloor\log_2 c\rfloor+1$?

Any answer or comment will be greatly appreciated!

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    $\begingroup$ I don't have any deep thoughts about it, but just an observation that if the class is very large, then the coclass is small and then the derived length is also small. So you won't have equality in this case. $\endgroup$
    – Yiftach Barnea
    Mar 22, 2018 at 23:40
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    $\begingroup$ Another stupid observation any group of class 2 satisfies this condition. Thus, as observed by the answer below, a classification is unlikely. Maybe a more sensible question is to ask for a group of order $p^n$ what is the maximal class for which such equality can still be achieved? $\endgroup$
    – Yiftach Barnea
    Mar 23, 2018 at 8:02

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My guess is "no", at least as regards a classification. A check with GAP shows that 30,591 of the 34,297 groups of order $5^7$ satisfy this equality; that 7,882 of the 9,310 groups of order $3^7$ satisfy the equality; and that 53,499 of the 56,092 groups of order $2^8$ satisfy the equality.

Here's an easy bit of GAP code to check this:

glist:=AllSmallGroups(Size,5^7,G->DerivedLength(G) = 1 + LogInt( NilpotencyClassOfGroup(G) , 2));; NrSmallGroups(5^7); Length(glist);

Partial information may be attainable, such as constraining various aspects of their character theory, but off-hand I'm not sure. This paper of Glasby's seems to contain all the references needed for the original derivations of the inequality in question (he references [1],[2], and [4] in his bibliography).

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  • $\begingroup$ Dear zibadawa timmy, thank you for your answer. $\endgroup$
    – sebastian
    Mar 25, 2018 at 17:19

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