I am attempting to solve questions on groups akin to:
Let $a=(12)(34)$ and $b=(24)$ elements of symmetric group $S_4.$ Subgroup $G=\langle a,b\rangle$ generated by $a,b$ is isomorphic to ? Choices are
a) Dihedral group of order $4$
b) Dihedral group of order $8$
c) $S_3$
d) $A_4$
My attempt: $a^2=b^2=(ab)^4=1$; So,
- it can't be a) since a) is generated by $a^4=1, b^2=1$.
- It can't be b) since $D_8$ is generated by $a^8=1, b^2=1$;
- It can't be c) since there is an element of order $4.$
- And it can't be d) since $ab$ is an odd permutation.
What am I missing here?
Also another example: Let $D_{32}=\langle r,s\mid r^{16}=s^2=1,rs=sr^{−1}\rangle$ and $H=\langle r^9,s \rangle$ is subgroup of $D_{32}$. Then $H$ is isomorphic to ? Can somebody give me a hint as to how to go about solving problems of this type?