# Using the 6 sets of sets of 5 disjoint 2-cycles and the 3 sets of 5 pairwise disjoint triangles to exhibit an outer automorphism of $S_6$.

As a follow-up to the construction of a graph that $S_6$ acts on in Constructing a graph based on numbers of vertices, incident edges, and incident triangles, I am now specifically looking at the relationship between the 6 sets of sets of 5 disjoint 2-cycles (ie $\{(ab),(ac),(ad),(ae),(af)\}$ to the 3 sets of 5 pairwise disjoint 2-cycle triangles (ie $\{\{(ab),(cd),(ef)\},\{(ac),(be),(df)\},\{(ad),(bf),(ce)\},\{((ae),(bd),(cf)\},\{(af),(bc),(de)\}\}$. I think it's important that $\frac{6}{3} =2$ but I am still a bit lost on how to construct an outer automorphism from this info. I would appreciate it if someone could steer me in the right direction.

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It's difficult to try and figure out what you are expected to do from isolated individual queries. What was the point of the construction of the graph with 15 vertices? I assumed that you be using that in some way to help you construct the outer automorphism. –  Derek Holt Oct 24 '12 at 7:53

Consider sets $A$ consisting of 5 sets of three disjoint transpositions, such that each transposition appears exactly once.
To find these sets, it makes sense to look at all transpositions having a common element. For example, look at $T = \{(12),(13),(14),(15),(16)\}$. In a set $A$, all these transpositions must be present. Furthermore, since $A$ has $\binom{6}{2}/3 = 15/3 = 5$ elements, and no element of $A$ can contain two transpositions of $T$, each element of $T$ is contained in exactly one element of $A$. From this information, it is not too hard to construct all suitable sets $A$.
There are 6 such sets: $$\{(12)(34)(56),(13)(25)(46),(14)(26)(35),(15)(24)(36),(16)(23)(45)\} \\ \{(12)(34)(56),(13)(26)(45),(14)(25)(36),(15)(23)(46),(16)(24)(35)\} \\ \{(12)(35)(46),(13)(24)(56),(14)(25)(36),(15)(26)(34),(16)(23)(45)\} \\ \{(12)(35)(46),(13)(26)(54),(14)(23)(56),(15)(24)(36),(16)(25)(34)\} \\ \{(12)(36)(45),(13)(24)(56),(14)(26)(35),(15)(23)(46),(16)(25)(34)\} \\ \{(12)(36)(45),(13)(25)(46),(14)(23)(56),(15)(26)(34),(16)(24)(35)\}$$
Now any $\sigma\in S_6$ permutes this $6$-element set by applying it to all the numbers in the cycles simultaneously. Show that this gives rise to an outer automorphism $S_6\to S_6$.