Let $G= ${$e, r, r^2, . . . , r^{23}, s, sr, sr^2, . . . , sr^{23} $} be the dihedral group with $48$ elements.
(a) Compute which elements of $G$ have order $3$.
(b) Determine if $H =$ {$r^i$ with $i$ even, $sr^j$ with $j$ odd} is a subgroup of $G$. Hint: perform the required calculations with types of element $sr^l$ and $sr^l$.
(a) We know that if H is a subgroup of G then $|H|\mid|G|$
From this, we get that the elements of order 3 are $r^8$ and $r^{16}$
since $(r^8)^3 = r^{24} =1 \implies|r^8|=3 $
and $(r^{16})^3 = r^{48} = r^{24} r^{24} =1 \implies |r^{16}|=3$
(b) $H=$ {$r^2,r^4,r^6,...,r^{22},sr^1,sr^3,sr^5,....sr^{23}$}
H is a subgroup of G if $H \ne \emptyset$ and for every $x,y \in H$ $ xy^{-1} \in G$.
Then since $H \ne \emptyset$ and $sr = r^{-1}s \in H$ it follows that H is a sungroup of G.
Am I right?