Paper 3, Section II, D

Groups | Part IA, 2021

(a) Let AA be an abelian group (not necessarily finite). We define the generalised dihedral group to be the set of pairs

D(A)={(a,ε):aA,ε=±1}D(A)=\{(a, \varepsilon): a \in A, \varepsilon=\pm 1\}

with multiplication given by

(a,ε)(b,η)=(abε,εη)(a, \varepsilon)(b, \eta)=\left(a b^{\varepsilon}, \varepsilon \eta\right)

The identity is (e,1)(e, 1) and the inverse of (a,ε)(a, \varepsilon) is (aε,ε)\left(a^{-\varepsilon}, \varepsilon\right). You may assume that this multiplication defines a group operation on D(A)D(A).

(i) Identify AA with the set of all pairs in which ε=+1\varepsilon=+1. Show that AA is a subgroup of D(A)D(A). By considering the index of AA in D(A)D(A), or otherwise, show that AA is a normal subgroup of D(A)D(A).

(ii) Show that every element of D(A)D(A) not in AA has order 2 . Show that D(A)D(A) is abelian if and only if a2=ea^{2}=e for all aAa \in A. If D(A)D(A) is non-abelian, what is the centre of D(A)?D(A) ? Justify your answer.

(b) Let O(2)\mathrm{O}(2) denote the group of 2×22 \times 2 orthogonal matrices. Show that all elements of O(2)\mathrm{O}(2) have determinant 1 or 1-1. Show that every element of SO(2)\mathrm{SO}(2) is a rotation. Let J=(1001)J=\left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right). Show that O(2)\mathrm{O}(2) decomposes as a union SO(2)SO(2)J\mathrm{SO}(2) \cup \operatorname{SO}(2) J.

[You may assume standard properties of determinants.]

(c) Let BB be the (abelian) group {zC:z=1}\{z \in \mathbb{C}:|z|=1\}, with multiplication of complex numbers as the group operation. Write down, without proof, isomorphisms SO(2)BR/Z\mathrm{SO}(2) \cong B \cong \mathbb{R} / \mathbb{Z} where R\mathbb{R} denotes the additive group of real numbers and Z\mathbb{Z} the subgroup of integers. Deduce that O(2)D(B)\mathrm{O}(2) \cong D(B), the generalised dihedral group defined in part (a).

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