Subgroup of free group

Question

Let $A=\{a,b\}$ and $FG(A)$ be the free group over $A$. Let $H=<a^2,b>$ and $K=<a,b^3>$ be subgroup of $FG(A)$. Are the elements $a^{2n+1}b^{3m+1}$ ($n,m\in \mathbb{N}$) in $HK$?

Answer 1

Consider the quotient $G$ of $FG(A)$ obtained by imposing the relations $a^2 = 1$ and $b^3 = 1$. Denote by $\bar{a}$ and $\bar{b}$ the images of $a$ and $b$ in the quotient group $G$, and similarly denote by $\bar{H}$ and $\bar{K}$ the homomorphic images of $H$ and $K$ respectively (in the quotient group $G$). Then $\bar{H}$ is then generated by $\bar{b}$ alone, and $\bar{K}$ is generated by $\bar{a}$ alone. It is then clear that $\bar{a}\bar{b}$ does not belong to $\bar{H}\bar{K}$. Hence $ab$ does not belong to $HK$.

Remark: of course $G$ is isomorphic to the free product of $\mathbb{Z}_2$ with $\mathbb{Z}_3$.