Let $G$ be a finite $p$-group for some prime $p$, and let $\rho: G \to \text{Aut}(A)$ a faithful representation of $G$ for some finite abelian $p$-group $A$ (which exists because, for example, we may embed $G$ in a symmetric group $S_n$ and then embed the latter in $\text{GL}_n(\mathbb{F}_p)$ as the group of permutation matrices). It can be shown that for any $\phi \in \text{Aut}(A)$ whose order is a power of $p$, $\phi-\text{id}_A$ is nilpotent, and in particular this holds for all elements of $\rho(G)$; therefore, we can define the nilpotency class $c_\rho(G)$ as the smallest $k$ such that $(\rho(g)-\text{id}_A)^k=0$ for all $g \in G$. I wonder how these numbers are related to the ordinary nilpotency class $c$ of $G$, in particular:
- Do we always have $c_\rho(G) \ge c$ ?
- If yes, when can we find a $\rho$ that achieves the equality? (For example, it is obviously impossible if $c=1$, but maybe it is the only exception).
I actually don't see an evident connection between the two notions, apart from the fact that while doing some computations in some nonabelian $p$-groups, and in particular while computing the $p$-th power of a generic element in terms of a fixed generating set, I often see binomials of the form $\binom{p}{k}$ appearing, where $k$ is at most the nilpotency class (similarly to when we compute the $p$-th power of a matrix of the form $I+N$ where $N$ is nilpotent).
Is there any reference about this?
(In this I take $A$ to be elementary abelian, so we are just considering representations of $G$ over $\mathbb F_p$.)
There is no connection. On the one hand, if $G=\langle x\rangle$ is a cyclic $p$-group then a representation of $G$ sends $x$ to a matrix of order $|G|$. Such an example is a single Jordan block of order $|G|$, and this has nilpotency class $|G|$. So $c_\rho(G)=|G|$ and $c=1$.
On the other hand, let $G$ be a $p$-group of exponent $p$. In any representation of $G$, the Jordan blocks of the elements have size at most $p$, so $c_\rho(G)\leq p$. On the other hand, $c$ can be much larger than $p$. For example, there is a $2$-generator group of exponent $5$ of nilpotency class $12$, and there is a $3$-generator one of class $17$. In general, the nilpotency class of a $p$-group of exponent $p$ for $p\geq 5$ is unbounded.
In general $c_\rho(G)$ is bounded above by the exponent $e$ of $G$ (and also bounded below by $e/p+1$), which bears little relation to the nilpotency class.