Paper 3, Section I, D

Groups | Part IA, 2016

Let GG be a group, and let HH be a subgroup of GG. Show that the following are equivalent.

(i) a1b1abHa^{-1} b^{-1} a b \in H for all a,bGa, b \in G.

(ii) HH is a normal subgroup of GG and G/HG / H is abelian.

Hence find all abelian quotient groups of the dihedral group D10D_{10} of order 10 .

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