= Solution
Write $r=g_1$. The matrix $D(s)$ is a reflection by part (a). Reflection reverses a rotation angle, giving $D(s)D(r)D(s)=D(r)^{-1}$. Faithfulness lifts this matrix identity back to the group:
$$
srs=r^{-1},\qquad r^p=s^2=I.
$$
Moving each $s$ past a rotation reduces every word to $r^i$ or $r^is$, $0\leq i<p$. These $2p$ elements are distinct: the rotations are distinct by the order of $r$, and reflections have determinant $-1$, so cannot equal a rotation. Therefore
$$
\boxed{|G|=2p,\qquad G\cong D_{2p}.}
$$
Here the subscript denotes group order. This is the reflection case of <finite subgroups of the plane orthogonal group>.
Every $r^is$ has square $r^isr^is=r^ir^{-i}=I$, so there are $p$ reflection subgroups of order two. Within the rotations, the even-$p$ assumption from part (b) supplies the additional half-turn $r^{p/2}$. Distinct involutions generate distinct two-element subgroups. Thus
$$
\boxed{\text{Number of order-two cyclic subgroups}=p+1\quad(p\text{ even}).}
$$
For comparison, without the evenness requirement the corresponding odd-$p$ count would be $p$. The even case agrees with <involutions in an even dihedral group>.
For $p=4$, take
$$
R=D(r)=\begin{pmatrix}0&-1\\1&0\end{pmatrix},\qquad S=D(s)=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
$$
Direct multiplication gives $SRS=\begin{pmatrix}0&1\\-1&0\end{pmatrix}=R^3$, so $srs=r^3=g_3$. The product rules are
$$
r^ir^j=r^{i+j},\quad r^i(r^js)=r^{i+j}s,\quad(r^is)r^j=r^{i-j}s,\quad(r^is)(r^js)=r^{i-j},
$$
with exponents modulo four. They yield the complete <multiplication table>, where the row element multiplies the column element on the left:
$$
\begin{array}{c|rrrrrrrr}
&I&r&r^2&r^3&s&rs&r^2s&r^3s\\\hline
I&I&r&r^2&r^3&s&rs&r^2s&r^3s\\
r&r&r^2&r^3&I&rs&r^2s&r^3s&s\\
r^2&r^2&r^3&I&r&r^2s&r^3s&s&rs\\
r^3&r^3&I&r&r^2&r^3s&s&rs&r^2s\\
s&s&r^3s&r^2s&rs&I&r^3&r^2&r\\
rs&rs&s&r^3s&r^2s&r&I&r^3&r^2\\
r^2s&r^2s&rs&s&r^3s&r^2&r&I&r^3\\
r^3s&r^3s&r^2s&rs&s&r^3&r^2&r&I\\
\end{array}
$$
This is \b[the symmetry group of a square], the <dihedral group> of order eight, often written $D_4$ when its subscript counts vertices instead. Its five order-two subgroups are generated by $r^2,s,rs,r^2s,r^3s$.
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