Explanation of group theoretic notation (lower central exponent $p$ series) Apologies for the slightly trivial question. I'm learning about the lower exponent-$p$ central series and I do not really understand the notation.
So it is defined as follows for a group $G$ and a prime number $p$: $L_0(G)=G$ and $L_i=[L_{i-1}(G),G]L_{i-1}(G)^p$ for $i\geq 1$. My question is: how is $H^q$ defined for a group $H$ and a natural number $q$?
My first thought was that this was the direct product, but if $L_i(G)\geq L_{i+1}(G)$ then this cannot be the case. So I thought that it might be the group generated by the $q$-fold products of elements of $H$, but then this would be the same as $H$?
Thanks for clarifying.

As you note, if the meaning was similar to the ideal-theoretic one and you define $H^q$ to be the set (or subgroup generated by the set) of $q$-fold products of elements of $H$, then $H^q=H$ would hold whenever $H$ is a subgroup, since you could take $q-1$ of the factors to be the identity. That is not the intended meaning here.
Rather, given a group $G$ and a positive number $n$, let
$$G^{\{n\}} = \{g^n\mid g\in G\}$$
be the set of $n$th powers of elements of $G$, and let
$$G^n = \langle G^{\{n\}}\rangle$$
be the subgroup this set generates. It is easy to verify that $G^n$ is a normal subgroup (in fact, it is characteristic; in fact, it is fully invariant; in fact, it is verbal; there is a discussion of this hierarchy of subgroups inside this answer).
That is almost certainly the intended meaning here, since $G/G^n$ is the largest quotient of $G$ that has exponent $n$. With your definition, the $[L_{i-1}(G),G]$ factor ensures you have a central series, and the $L_{i-1}(G)^p$ factor ensures that the quotient is of exponent $p$, and any central series $N_k\leq\cdots\leq N_0=G$ for which the quotients have exponent $p$ will lie "above" this series the same way that any central series lies "above" the lower central series and below the upper central series, thus explaining the name.
