Find a cyclic subgroup of order 4 in $A_8$

Question

I need to find a subgroup of $A_8$ of order 4 that is cyclic. Can someone please help me with this?

Thanks in advance!

Answer 1

You need to find an element of $A_8$ of order $4$. What permutations of order $4$ do you know?

Answer 2

Thought: Of course a 4-cycle has order four, but sadly a 4-cycle is an odd permutation. Could you multiply it by something so the result is an even permutation but the order is still 4?

Answer 3

Here are some details on paw88789's answer:

• $(1234)$ has order $4$ but is an odd permutation.

• $(1234)(ab)$ is an even permutation that will have order $4$ if $(1234)$ and $(ab)$ commute.

Answer 4

Let $\alpha = (1234) (56)$

Recall the thm Order of a Permutation: "The order of a permutation of a finite ser written in a disjoint cycle form is the least common multiple of the lengths of the cycles."

Thus:

$|\alpha| = |(1234)(56)|$

$|\alpha| = lcm(4,2)$

$|\alpha| = 4$

Hence we can see that $|<\alpha>| = |\alpha| = 4$