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Is this statement True or False-

If F is a free group with basis {$x,y$} and H is the subgroup generated by {$x^2,y^2,xy,yx$} then H is a free group of rank $3$.

What should be my approach to solve it. I do not know how to proceed.

Bhaskar Vashishth
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1 Answers1

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Any subgroup of a free group is free. To compute the rank of $H$, note that $H$ is the kernel of the map $F \to \mathbb{Z}_2$ given by $x \to 1, y \to 1$, and use the Nielsen-Schreier theorem. Alternatively, show that the abelianization of $H$ is isomorphic to $\mathbb{Z}^3$ by considering it as the kernel above (and note that $yx = y^2(xy)^{-1}x^2$).

anomaly
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  • To see it as kernel of some map, how to get that intution, it doesnot always have to be easy... right? – Bhaskar Vashishth Feb 05 '15 at 02:31
  • In this case, the idea is the given generators of $H$ are exactly the words of length $2$ in $F$, and so $H$ consists exactly of those words in $F$ of even length. (This isn't quite immediate; you have to check, for example, that cancellation (e.g., $xx^{-1} \to 1$) preserves the parity of the length.) The length function $F\to \mathbb{Z}$ is given by $x \to 1, y \to 1$, so we can reduce mod $2$ to get the definition of $H$ that I mentioned. – anomaly Feb 05 '15 at 02:36