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Let $G$ be a finite group generated by some finite set $S = \{g_1, g_2, ...\} \subseteq G$. Let $h \in G$ be some element. Let the function $c_n: G \rightarrow \mathbb{N}$ be defined that $c_n(h)$ is the number of ways of writing $h$ as a product of $n$ elements from $S$.

Has this function been studied in the literature? Are there known results of the forms this function can take and growth rates it exhibits, based on the choice of $G, S, h$?

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    $\begingroup$ $c_n(h)$ is counting the number of words in a regular language and so has a rational generating function and either polynomial or exponential growth. Your automaton is the Cayley graph of G with initial state 1 and accepting state h $\endgroup$
    – Benjamin Steinberg
    Commented Aug 21, 2022 at 17:44
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    $\begingroup$ Here is a related question about the spectra of Cayley graphs of finite groups. mathoverflow.net/q/18212/22277 $\endgroup$
    – Joseph Van Name
    Commented Aug 21, 2022 at 18:32

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