Sorry for posting a second question on this topic, abstract algebra is taking a bit longer to get my head around.
I'm trying to work out if this is cyclic or not, and find all generators or show no generator exists.
$\langle(12)(34)(56),(145)(236)\rangle \leq S_6$
Just in case there's differences in notation, that's just the subgroup generated by the elements $(12)(13)(56)$ and $(145)(236)$ of $S_6$. The problem is, my notes only show one example like this, but both permutations were even, and it was in $S_4$ so they just showed that you could generate $A_4$ (by literally showing you could make every even permutation) using the two elements. Given how one of the elements here is odd, and we're dealing with $S_6$ it doesn't look like a good method.
So far, I've noted that, letting $\sigma = (12)(34)(56)$ and $\tau = (145)(236)$ we get that $o(\sigma) = 2$ and $o(\tau) = 3$
Also, that $\sigma \tau = (135246) = \tau \sigma$, $(\sigma \tau)^2 = (\tau \sigma)^2 = (154)(326) = \tau^{-1}$
Lastly that $(\sigma \tau)^3 = (\tau \sigma)^3 = (12)(34)(56) = \sigma = \sigma^{-1}$
I get a bit stuck as to where to go at this point, although I can't see how it's even possible this is cyclic. Thanks again, this place has already been a massive help!
Best Answer
The most important observation is that $\sigma$ and $\tau$ commute; without that you could never have that they generate a cyclic subgroup. But once this is checked, you know they generate a commutative subgroup; you can ignore the rest of $S_6$ and apply the theory of Abelian groups. In an Abelian group, two elements $a,b$ with relatively prime orders $p,q$ always have the property that $ab$ has order $pq$: the relative primality ensures that $\langle a\rangle\cap\langle b\rangle=\{e\}$, so $(ab)^i=a^ib^i=e$ implies $a^i=e=b^i$, hence $\operatorname{lcm}(p,q)=pq\mid i$. In your case you have elements $\sigma,\tau$ of relatively prime orders $2,3$, so $\sigma\tau$ has order $6$ and generates $\langle\sigma,\tau\rangle$, which is therefore cyclic.
The argument of course shows that whenever in any group you have commuting elements of relatively prime orders, they generate a cyclic group (you can even have more than two generators to start with, provided their orders are pairwise relatively prime; just iterate the argument).