Paper 3, Section I, D

Groups | Part IA, 2015

Say that a group is dihedral if it has two generators xx and yy, such that xx has order nn (greater than or equal to 2 and possibly infinite), yy has order 2 , and yxy1=x1y x y^{-1}=x^{-1}. In particular the groups C2C_{2} and C2×C2C_{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.

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