Paper 3, Section I, D

Say that a group is dihedral if it has two generators $x$ and $y$, such that $x$ has order $n$ (greater than or equal to 2 and possibly infinite), $y$ has order 2 , and $y x y^{-1}=x^{-1}$. In particular the groups $C_{2}$ and $C_{2} \times C_{2}$ are regarded as dihedral groups. Prove that:

(i) any dihedral group can be generated by two elements of order 2 ;

(ii) any group generated by two elements of order 2 is dihedral; and

(iii) any non-trivial quotient group of a dihedral group is dihedral.

*Typos? Please submit corrections to this page on GitHub.*