How to identify a cyclical subgroup?

Question

I've been given a piece of homework and it asks which of the two subgroups are cyclic:

a) $⟨1,2,3⟩, id, ⟨1,3,2⟩$ on S₁₁

b) $⟨1,2⟩, ⟨3,4⟩, id, ⟨1,2⟩⟨3,4⟩$ on S₅

Apparently a) is cyclic and b) is not but how do I work this out?

I'm not particularly familiar with notation for groups but in the long-term, it would be useful to understand how to identify future cyclical subgroups.

Ultimately, I will be writing a code to determine whether an input is a cyclical subgroup or not so I'm really look for a somewhat formulaic way to determine this.

Answer 1

Here a) is isomorphic to $$\Bbb Z_3\cong\langle x\mid x^3\rangle,$$ so is cyclic with either nonidentity element as a generator (mapping to $x$), whereas b) is isomorphic to the Klein four group $$\Bbb Z_2\times\Bbb Z_2\cong\langle a, b\mid a^2, b^2, ab=ba\rangle$$ under the map given by $(12)\mapsto a, (34)\mapsto b,$ so is thus not cyclic.

The notation I'm using here is that of presentations of groups.

One general method to decide whether a given group is cyclic is to take a nonidentity element then multiply it by itself repeatedly. If you reach every element of the group, then it is cyclic; if not, try another element; and if the group is finite, this process will halt.