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$$ \begin{array}{|l|c|c|} \hline \text{cycle structure} & \text{number of permutations} & \text{order} \\ \hline 6 & 120 & 6 \\ 5+1 & 144 & 5 \\ 4+2 & 90 & 4 \\ 4+1+1 & 90 & 4 \\ 3+3 & 40 & 3 \\ 3 + 2 + 1 & 120 & 6 \\ 3 + 1 + 1 + 1 & 40 & 3 \\ 2 + 2 + 2 & 15 & 2 \\ 2 + 2 + 1 + 1 & 45 & 2 \\ 2 + 1 + 1 + 1 + 1 & 15 & 2 \\ 1 + 1 + 1 + 1 + 1 + 1 & 1 & 1 \\ \hline \end{array} $$

In trying to understand outer automorphisms of $S_6$ at the most concrete level of arithmetic, it appears to me that the $15$ generators whose cycle structure is $2+1+1+1+1$ get mapped to the $15$ permutations whose structure is $2+2+2$ (which also generate the group). And the $40$ permutations whose structure is $3+1+1+1$ similarly get interchanged with the $40$ whose structure is $3+3$; likewise the $120$ whose structure is $3+2+1$ with the $120$ whose structure is $6$.

But apparently the set of $90$ whose structure is $4+2$ is invariant, as is the set of $90$ whose structure is $4+1+1$.

  • Is this in accord with the experience of those who've actually thought about this?
  • Does some web page somewhere walk through this? I would think the routine arithmetic would have been figured out.
Michael Hardy
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    I take it you've seen http://en.wikipedia.org/wiki/Automorphisms_of_the_symmetric_and_alternating_groups already. Maybe http://polyomino.f2s.com/david/haskell/outers6.html or http://math.ucr.edu/home/baez/six.html ? – Matthew Towers Mar 22 '12 at 20:26
  • Perhaps http://groupprops.subwiki.org/wiki/S6#Up_to_automorphism this is of some help? – MJD Mar 22 '12 at 20:42
  • I don't know that I've seen the one from John Baez before. He quotes Chris Henrich: "a permutation of 123456 induces a permutation of ABCDEF. A two-cycle such as (12) induces a product of three disjoint two-cycles such as (AB)(CD)(EF)". So the set of six things getting permuted is different in the two cases. It had irritated me that there seems to be no natural choice of correspondence between the $(2+1+1+1+1)$-structured permutations and the $(2+2+2)$-structured ones. This takes that into account rather neatly. – Michael Hardy Mar 22 '12 at 22:34
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    BTW, it is clear that $4+2$ and $4+1+1$ are preserved, because the first is in $A_6$ and the second is not. – Derek Holt Mar 23 '12 at 13:58
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    @Derek: I think it's worth indicating why $A_6$ is invariant under outer automorphisms (namely it's the commutator subgroup). – Qiaochu Yuan Mar 23 '12 at 18:21
  • I just wrote about it here : http://settheory.net/Out(S6) – user27887 Mar 16 '18 at 19:15

1 Answers1

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I think the best source for understanding the outer automorphisms of $S_6$ is the following paper:

Combinatorial Structure of the automorphism group of $\mathbf{S_6}$ by T.Y. Lam and David B. Leep, Expositiones Mathematicae 11 (1993), no. 4, pp. 289-903.

It describes exactly how to think about the exchange map between $2$-cycles and "pseudo-transpositions" $(ab)(cd)(ef)$, and it gives nice ways to interpret them combinatorially. I highly recomment it.

Arturo Magidin
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    I think the best way to see where this outer automorphism comes from is to find a copy of $S_5$ in $S_6$ which is not a point stabilizer. The coset action then induces a map from $S_6$ to itself, which is the outer automorphism. It is also not too hard to show for other values of $n$, every $S_{n-1}\subset S_n$ is a point stabilizer. –  Mar 23 '12 at 02:48
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    A nice way to start looking for a strange embedding of S5 in S6 is with the dodecahedron: the rotation group acts faithfully on the set of unordered pairs of opposite faces, so embeds into S6 in a way which is clearly fixed-point-free. But the rotation group is A5. You can see easily that 3-cycles go to double 3-cycles if you have a model dodecahedron at hand. – Matthew Towers Mar 23 '12 at 10:49
  • OK, now I've got the article and I'm starting to look it over...... – Michael Hardy Mar 24 '12 at 04:20
  • @MichaelHardy: I hope you find it useful; personally, I really like the way Lam writes, and I really liked this paper. I guess it's possible others may disagree with me on either count (or both!)... – Arturo Magidin Mar 25 '12 at 21:58
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    Surely this is not a 614 page paper? – Michael Joyce Oct 20 '12 at 20:45
  • I believe the page numbers should be 289 to 308. – Gerry Myerson Apr 10 '15 at 03:29
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    Any idea where I can get a copy of the article? The journal's website only seems to provide the articles from 2001 on... – Nicolas Miari Aug 22 '16 at 06:40