Consider the free group of rank $2$, denoted by $\mathbb{F}(x,y)$.
I want to show that the subgroup generated by $\{x^{2},xy,y^{2}\}$ is free group of rank $3$.
How should I approach this problem ? Any hints.
2026-03-26 19:02:53.1774551773
Subgroup in Free group of rank $2$ is isomorphic to a free group of rank $3$
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When dealing with free groups, I like to think about finite graphs. In your situation, notice that you have the following covering of graphs (which is $\pi_1$-injective in particular):