= Diameter of a finite group with respect to a generating set
{title2=$\operatorname{diam}_S(G)$}
For a <finite group> $G$ with a finite symmetric <generating set> $S$, its diameter is the maximum <word metric> distance between two elements, equivalently the <diameter> of its <Cayley graph>. Put
$$
B_S(n)=\bigcup_{k=0}^{n}S^k=(S\cup\{1\})^n.
$$
Then
$$
\operatorname{diam}_S(G)=\min\{n:B_S(n)=G\}.
$$
Translation invariance reduces the maximum pairwise distance to the maximum distance from the identity, which proves the equality.
If $1\in S$, one may replace $B_S(n)$ by $S^n$. Otherwise exact powers may never equal $G$: for $G=C_2=\{1,a\}$ and $S=\{a\}$, even powers are $\{1\}$ and odd powers are $\{a\}$, while the actual <Cayley graph> has diameter one. This is why the ball convention matters. The <trivial diameter bound for a finite group> gives $\operatorname{diam}_S(G)\leq|G|-1$.
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