= Conjugates of one free generator freely generate
If $a,b$ freely generate, then $a^iba^{-i}$ for any distinct integer indices freely generate their subgroup. Group a reduced word in these new generators into blocks with nonzero exponents and different successive indices. Expansion produces alternating nonzero powers of $b$ and interior powers $a^{i_{j+1}-i_j}$, so it cannot freely reduce to the empty word. Consequently every finite-rank free group embeds in a rank-two free group; free rank is not monotone under subgroup inclusion.
Back to article page
