Let $G$ be a finite $p$-group of maximal nilpotency class that is not cyclic of order $p^2$. Then $G$ is $2$-generated, say $G=\langle a,b\rangle$. Is there a classification in the case when $a^p=b^p=1$? Comments appreciated.
1 Answer
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The classification of such groups is as difficult as the classification of all $p$-groups of maximal class. Note that, for the latter problem, beside the cases where $p=2,3$ that were settled by Blackburn in his Acta Math. paper (1958), only a little progress is made in the case $p=5$.
To see why, suppose that $G$ is a $p$-group of maximal class and order $p^n$; let $G_i$ denote the $i$th term of the lower central series of $G$, and let $K_i:=C_G(G_i/G_{i+2})$ for $i=2,\dotsc,n-2$. Observe that $G_i/G_{i+2}$ has order $p^2$ and so $K_i$ is a maximal subgroup of $G$. The key fact is the following:
For every $x\in G$ which is not in any of the $K_i$'s we have $x^p\in G_{n-1}$.
Indeed, for such a $x$ we have $x^p\in G_i\setminus G_{i+1}$ for some $i\geq 2$. If the above claim is false, then we should have $i\leq n-2$, and hence the image of $x^p$ in $G_i/G_{i+1}$ is a generator of this factor (because $G_i/G_{i+1}$ has order $p$); as $x$ commutes with $x^p$ we have $$[x,G_i]=[x,\langle x^p\rangle G_{i+1}]=[x,G_{i+1}]\leq G_{i+2},$$ so $x\in K_i$, a contradiction.
Next, note that in most cases (the non-degenerate cases if you wish) all the $K_i$'s are equal—which is equivalent to saying that $G$ has a positive degree of commutativity. This holds for example for $G$ of order $p^n$ with $n> p+1$. For such a $G$, if we pick any two elements $a,b$ outside $K_1$ that generate $G/G_2$, then, by virtue of the above result, $G/G_{n-1}$ is a $p$-group of maximal class that is generated by two elements of order $p$ (namely, by $a$ and $b$ modulo $G_{n-1}$).
To summarize, a classification of $p$-groups of maximal class having generators of order $p$ yields a classification of all $p$-groups of maximal class up to a very good approximation (i.e., up to a central extension with kernel of order $p$). Using the language of "coclass trees", the $p$-groups you are considering determine to a large extent the structure of the graph of $p$-groups of maximal class: here the vertices are the $p$-groups of maximal class (up to isomorphism), and there is a directed edge $G\to H$ if there is an epimorphism from $H$ to $G$ with kernel of order $p$. The structure of such graphs is still far from being understood despite the recent breakthroughs made by the coclass theorists (cf. e.g., Dietrich - A new pattern in the graph of $p$-groups of maximal class). Wish that this helps!
