What properties the permutations $\sigma=(1 2)(3 4 5)$ and $\tau=(1 2 3 4 5 6)$ in $S_6$ satisfies?

Question

Let $\sigma=(1 2)(3 4 5)$ and $\tau=(1 2 3 4 5 6)$ in $S_6$,

1. The subgroups $<\sigma>$ and $<\tau>$ are isomorphic to each other.
2. $\sigma$ and $\tau$ are conjugate in $S_6$.
3. $<\sigma>\cap<\tau>$ is a trivial group.
4. $\sigma$ and $\tau$ commute.

$2$ and $4$ are not true as I have checked. But I am unable to conclude $1$ and $3$. Help me out. Thanks.

Answer 1

Note that $\sigma$ has order 6 since the lcm of its cycle lengths in its cycle decomposition is 6. (Alternatively just check). Then $\langle \sigma\rangle \cong Z_6\cong \langle\tau\rangle$, so you just need to check whether or not they have trivial intersection. One method is to just write them out. But alternatively, note that $\sigma$ fixes 6, so that all elements of $\langle\sigma\rangle$ fix 6. However, if $\tau^n$ fixes six, then $n\equiv 0\pmod{6}$, since $\tau^n 6 \equiv 6+n \pmod{6}$. Therefore, the only element of $\langle\tau\rangle$ that can be in $\langle\sigma\rangle$ is the identity, so that their intersection is trivial.