Period of a particular finite group Let $G$ be a group fitting in the following exact sequence: $0 \to \mathbb{Z}/p \to G \to \mathbb{Z}/q^r \to 0.$ Here $q$ and $p$ are primes (not necessarily distinct). It is easy to check (by the general characterisation in terms of Sylow subgroups, for example) that $G$ has periodic cohomology [edit: unless $p=q$ and the sequence splits, I suppose. Assume the cohomology is cyclic.].
My question is, what is the period of the cohomology of this group? To be a bit more specific, what is the minimal length of a periodic resolution of $F$ as an $F[G]$ algebra, where $F$ is a field of characteristic $p$?
I would like the answer to be (a multiple of) $q^{r-l}$ where $q^l$ is the size of the kernel of the action of $\mathbb{Z}/q^r$ on $\mathbb{Z}/p,$ but I have no idea how to prove that (the inspiration of this guess comes from algebraic topology and seems too lengthy to me to explain here fruitfully).
 A: For $p\neq q$ it's precisely $2q^{r-l}$, unless $p=2$, in which case it's $1$.
My initial mental calculation used knowledge of the modular representation theory of groups like this, which seems too lengthy to me to explain fruitfully here (though you might like to check out "Brauer tree algebras"), but I can probably distill a more efficient proof if I think about it.
Edit: Here's a construction of a projective resolution of the trivial $FG$-module. I'll assume $p\neq q$ and $p\neq2$ (the other cases are easy).
Let $N=\mathbb{Z}/p$ and $Q=\mathbb{Z}/q^r$, let $g$ be a generator of $Q$, and let $J$ be the augmentation ideal of $FN$. Then there's a filtration
$$0<J^{p-1}<\dots<J^2<J<FN$$
by $FG$-modules. Then $g$ acts on the one-dimensional space $J/J^2$ by multiplication by a primitive $q^{r-l}$th root $\omega$ of $1$.
The short exact sequence
$$0\to J^2\to J\to J/J^2\to 0$$
of $FQ$-modules splits, since $p=\operatorname{char}(F)$ doesn't divide $|Q|$. So we can choose a generator $x$ of $J$ that is an eigenvector for the action of $g$ with eigenvalue $\omega$.
Then $FN=F[x]/(x^p)$.
As an $FN$-module, $F$ has a projective resolution of period $2$
$$\dots\to FN\stackrel{x^{p-1}}{\to}FN\stackrel{x}{\to}FN\to\dots\to FN\stackrel{x^{p-1}}{\to}FN\stackrel{x}{\to}FN\to F\to0.$$
The maps are not $Q$-equivariant. However, we can make them so by tensoring with suitable one-dimensional $FQ$-modules (with trivial $N$-action).
Let $V_i$ (for $i\in\mathbb{Z}/q^{r-l}\mathbb{Z}$) be the one-dimensional $FQ$-module on which $g$ acts by multiplication by $\omega^i$. Then
$$\dots\to FN\otimes V_{-p-1}\stackrel{x}{\to}FN\otimes V_{-p}\stackrel{x^{p-1}}{\to}FN\otimes V_{-1}\stackrel{x}{\to}FN\to F\to0,$$
and this resolution has period $2q^{r-l}$ unless $p=2$ (in which case $x^{p-1}=x$ and $r-l=0$), when it has period $1$.
