Let $G = \langle x, y \mid x^{7} = y^{3} = e,\; yxy^{−1} = x\rangle$. What is $|G|\,$?
Solution 1:
Claim: $G\cong C_7\times C_3$. Start with the map from the free group on two generators $F_2$ to $C_7\times C_3$ given by $x\mapsto(1,0)$ and $y\mapsto (0,1)$. Verify that $x^7$, $y^3$, and $yxy^{-1}x^{-1}$ are in the kernel of this map, so we have an induced quotient $f\colon G\to C_7\times C_3$. By Von Dyck's theorem (or directly) we know $f$ is surjective. Now argue that $|G|$ is at most $21$: every element can be written $x^my^n$ with $0\le m\le 6$ and $0\le n\le 2$.