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1) Let $F$ be the free group on the three generators $x,y,z$. For non-zero integers $r,s,t$

then CLAIM: the subgroup of $F$ generated by $x^r , y^s , z^t$ is freely generated by these elements.

2) Let $H$ be the subgroup of $F(\{x,y\})$ generated by $x^2 , y^2 , xy , yx$

then CLAIM: $H$ is not freely generated by these elements.

Can anyone help me?

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2) there is a nontivial relation among them: $$(x^2)^{-1}(xy) = x^{-1}y = (yx)^{-1}(y^2) $$ 1) Consider the homomorphism $\phi$ from $F$ to this subgroup $G:=\langle x^r,y^s,z^t\rangle$, mapping $x\mapsto x^r$, $y\mapsto y^s$, $z\mapsto z^t$, and conclude that $\ker\phi=\{1\}$, thus $im\phi\cong F$, free with those generators.

Anyway, the elements of $G$ look like words with strings of letters $X:=\overbrace{xx..xx}^r$, $Y:=\overbrace{yy..yy}^s$ and $Z=\overbrace{zz..zz}^t$ and their inverses $X^{-1}$, $Y^{-1}$, $Z^{-1}$, and $\phi^{-1}$ thus is just replacing the strings $X$ to $x$, etc.

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thank you so much! – Dexter Nov 2 '12 at 15:44

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