I'm trying show that a free group of rank $\ge2$ is non abelian, but I have no idea to prove this. Any suggestions?
2026-03-26 03:11:11.1774494671
Free groups of rank greater than 2
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Hint. Pick your favourite non-commutative group, say $G$, and two distinct elements $x$ and $y$ in the basis $X$. You can define a function $f$ from $X$ to $G$ that sends $x$ and $y$ to non-commuting elements in $G$, sending the rest of $X$ to (e.g.) $1\in G$. This extends to a homomorphism from $F$ to $G$. Use this to show that $x$ and $y$ do not commute.