I am trying to show the following:
I think I have tried to show the second part of. But I couldn't show the first part.
My proof for the second part is as follows:
2026-03-27 06:56:54.1774594614
Cardinality of ball in group with generating set
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You have shown that the free group attains this bound. You have two things left to prove:
a. That every group satisfies this bound. The hint in the comment is a nice way of doing this: consider the natural homomorphism $F(S)\rightarrow G$.
b. That no other group attains the bound. To do this, consider a non-empty word $W(S)$ which is equal to the identity in $G$ (e.g. $G\cong \langle S\mid A, B, ...\rangle$ and take $W(S)=A$). This corresponds to a loop in the Cayley graph. Can you see how the result follows?