Find eight elements commute with one element in S6 Find 8 elements that commute with (12)(34)(56). Do they form a subgroup of S6?
I actually have found 8 elements randomly, but I found ab is not in my 8 elements( for some a,b of my 8 elements), so I can conclude this is not a subgroup, right?
 A: The permutation 
$\sigma= \left(
\begin{array}{cccccc}
1 & 2 & 3 & 4 & 5 & 6\\
a_1 & a_2 & a_3 & a_4 & a_5& a_6
\end{array} \right)
$ commutes with the permutation $(1,2)(3,4)(5,6)$ if and only if the permutation $(a_1 a_2)(a_3a_4)(a_5a_6)$ is the same as the permutation $(12)(34)(56)$, that is, if and only if $(a_1 a_2)(a_3a_4)(a_5a_6)$ is a rearrangement of $(12)(34)(56)$. There are $2^3 \times 3! = 48$ such permutations. For instance, $\ \sigma= \left(
\begin{array}{cccccc}
1 & 2 & 3 & 4 & 5 & 6\\
3 & 4 & 2 & 1 & 5 & 6
\end{array} \right)
$ commutes with $(12)(34)(56)$ since $(34)(21)(56)= (12)(34)(56)$.
All these $48$ elements form a subgroup of $S_6$. Now, if you take just $8$ of them, they may not form a subgroup.
A: Think as if it was the Klein group in $S_4$ but in $S_6$, then: 
$$(13)(24)(56) \\
(13)(26)(45) \\
(13)(25)(46)\\
(14)(23)(56)\\
(14)(25)(36)\\
(14)(26)(35)\\
(12)(35)(46)\\
(12)(36)(45)\\$$
Are the elements you are looking for. And the set:
$$ A = \{ (13)(24)(56),
(13)(26)(45),
(13)(25)(46),
(14)(23)(56),
(14)(25)(36),
(14)(26)(35),
(12)(35)(46),
(12)(36)(45), (12)(34)(56), Id \}.$$ Is in fact a subgroup of $S_6$.
A: In any group $G$, for any specific element $g\in G$, the cyclic subgroup generated by $g$ always commutes with $g$.
Here the subgroup generated by three transpositions $(12), (34), (56)$ is in fact of order 8. (counting the identity element): they are 
$\{id, (12), (34), (56), (12)(34), (12)(56), (34)(56), (12)(34)(56)\}$.
A: In permutation group ,disjoint cycles commute and inverse pairs commute ( and a transposition is its own inverse )
So consider this set
{(5,6),(3,4),(1,2),(5,6)(1,2),(5,6)(3,4),(1,2)(3,4),(5,6)(3,4)(1,2),e=(1,2)(2,1)}
It can be easily seen that this group is closed under composition and inverses . 
So it forms the subgroup of S6.
I cannot figure out how many of the elements of S6 will commute though. 
