Paper 3, Section II, D

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

$D(A)=\{(a, \varepsilon): a \in A, \varepsilon=\pm 1\}$

with multiplication given by

$(a, \varepsilon)(b, \eta)=\left(a b^{\varepsilon}, \varepsilon \eta\right)$

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

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

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

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

[You may assume standard properties of determinants.]

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

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