Is there a subgroup of $S_7$ that's isomorphic to $\Bbb Z_2\times \Bbb Z_4$?

Question

I was given this question in a quiz today, and had a hard time answering it. My solution was very complicated and not elegant at all, which makes me think I might have been completely wrong, or at least missed an important part of the solution. So can anyone here give a simple answer? Can you prove or disprove the existence of such a subgroup? Thanks in advance.

Answer 1

The subgroup generated by $(12)$ and $(3456)$ is isomorphic to $\Bbb Z_2\times\Bbb Z_4$.

Answer 2

$\mathbb{Z}_2 \times \mathbb{Z}_4$ looks like two independent cyclic groups, one with order $2$ and the other with order $4$. Can this easily happen? Yes, because $2+4 < 7$.

Thus, let $\mathbb{Z}_2 \times \mathbb{Z}_4$ be represented as $\langle a, b \mid a^2 = b^4 = 1, ab = ba \rangle$.

Then map $a \mapsto (1 2)$ and $b \mapsto (4567)$. Here, I'm using cycle notation.