91Ó°ÊÓ

The binary tetrahedral group \(B\) is the group having the presentation $$ B=\left(r, s, t \mid r^{2}=s^{3}=t^{3}=r s t\right) . $$ (i) Prove that \(r s t \in Z(B)\) and that \(B /\langle r s t\rangle \cong A_{4}\) (the tetrahedral group). (ii) Prove that \(B\) has order 24 . (iii) Prove that \(B\) has no subgroup of order \(12 .\)

Prove that a free group of rank \(>1\) is not solvable.

Prove that a group is free if and only if it has the projective property.

Prove that every finite group is finitely presented.