is called a $p$-group if $|G|=p^n$ is a power of $p$ (in particular, we assume $G$ to be finite here). The *exponent* of a group is the least common multiple of the orders of its elements. A $p$-group $G$ is *elementary abelian* if it is abelian and of prime exponent (i.e., $G\cong \mathbb{F}_p^n$ for some $n$). The *Frattini-subgroup* $\Phi(G)$ of a group $G$ is the intersection of all maximal subgroups. If $G$ is a $p$-group then $\Phi(G)$ is the unique minimal normal subgroup of $G$ with elementary abelian quotient. The elements of $\Phi(G)$ are *non-generators* in $G$, that is, if $\{g_1,\dots,g_m\}$ generates $G$ then so does $\{g_1,\dots,g_m\}\setminus\Phi(G)$.

We define a $1$-fold commutator to be just a regular commutator and then a $c$-fold commutator is an element of the form $[x,z]$ with $x\in G$ and $z$ a $(c-1)$-fold commutator in $G$. A group $G$ is said to be *nilpotent* if there is some integer $c$ such that $c$-fold commutators are always trivial in $G$ and if this is the case then the *nilpotency class* of $G$ is the smallest such $c$. For example abelian groups are exactly the groups of nilpotency class $1$ and a group has nilpotency class $2$ if 