(a) Let be an abelian group (not necessarily finite). We define the generalised dihedral group to be the set of pairs
with multiplication given by
The identity is and the inverse of is . You may assume that this multiplication defines a group operation on .
(i) Identify with the set of all pairs in which . Show that is a subgroup of . By considering the index of in , or otherwise, show that is a normal subgroup of .
(ii) Show that every element of not in has order 2 . Show that is abelian if and only if for all . If is non-abelian, what is the centre of Justify your answer.
(b) Let denote the group of orthogonal matrices. Show that all elements of have determinant 1 or . Show that every element of is a rotation. Let . Show that decomposes as a union .
[You may assume standard properties of determinants.]
(c) Let be the (abelian) group , with multiplication of complex numbers as the group operation. Write down, without proof, isomorphisms where denotes the additive group of real numbers and the subgroup of integers. Deduce that , the generalised dihedral group defined in part (a).