Automorphisms of the symmetric and alternating groups
In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S_{6}, the symmetric group on 6 elements.
Summary
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Generic case
- Formally, is complete and the natural map is an isomorphism.
- : , and the outer automorphism is conjugation by an odd permutation.
- :
Exceptional cases
- :
- : , and is a semidirect product.
- : , and
The exceptional outer automorphism of S_{6}
Among symmetric groups, only S_{6} has a (non-trivial) outer automorphism, which one can call exceptional (in analogy with exceptional Lie algebras) or exotic. In fact, Out(S_{6}) = C_{2}.^{[2]}
This was discovered by Otto Hölder in 1895.^{[3]}^{[4]}
This also yields another outer automorphism of A_{6}, and this is the only exceptional outer automorphism of a finite simple group:^{[5]} for the infinite families of simple groups, there are formulas for the number of outer automorphisms, and the simple group of order 360, thought of as A_{6}, would be expected to have two outer automorphisms, not four. However, when A_{6} is viewed as PSL(2, 9) the outer automorphism group has the expected order. (For sporadic groups (not falling in an infinite family), the notion of exceptional outer automorphism is ill-defined, as there is no general formula.)
Construction
There are numerous constructions, listed in Template:Harv.
Note that as an outer automorphism, it's a class of automorphisms, well-determined only up to an inner automorphism, hence there is not a natural one to write down.
One method is:
- Construct an exotic map (embedding) S_{5} → S_{6}
- S_{6} acts by conjugation on the six conjugates of this subgroup;
- yielding a map S_{6} → S_{X}, where X is the set of conjugates. Identifying X with the numbers 1, ..., 6 (which depends on a choice of numbering of the conjugates, i.e., up to an element of S_{6} (an inner automorphism)) yields an outer automorphism S_{6} → S_{6}.
- This map is an outer automorphism, since a transposition doesn't map to a transposition, but inner automorphisms preserve cycle structure.
Throughout the following, one can work with the multiplication action on cosets or the conjugation action on conjugates.
To see that S_{6} has an outer automorphism, recall that homomorphisms from a group G to a symmetric group S_{n} are essentially the same as actions of G on a set of n elements, and the subgroup fixing a point is then a subgroup of index at most n in G. Conversely if we have a subgroup of index n in G, the action on the cosets gives a transitive action of G on n points, and therefore a homomorphism to S_{n}.
Exotic map S_{5} → S_{6}
{{safesubst:#invoke:anchor|main}} There is a subgroup (indeed, 6 conjugate subgroups) of S_{6} which are abstractly isomorphic to S_{5}, and transitive as subgroups of S_{6}.
Sylow 5-subgroups
Janusz and Rotman construct it thus:
- S_{5} acts transitively by conjugation on its 6 Sylow 5-subgroups, yielding an embedding S_{5} → S_{6} as a transitive subgroup of order 120. (The obvious map S_{n} → S_{n+1} fixes a point and thus isn't transitive.)
This follows from inspection of 5-cycles: each 5-cycle generates a group of order 5 (thus a Sylow subgroup), there are 5!/5 = 120/5 = 24 5-cycles, yielding 6 subgroups (as each subgroup also includes the identity), and S_{n} acts transitively by conjugation on cycles of a given class, hence transitively by conjugation on these subgroups.
One can also use the Sylow theorems, which imply transitivity.
PGL(2,5)
The projective linear group of dimension two over the finite field with five elements, PGL(2, 5), acts on the projective line over the field with five elements, P^{1}(F_{5}), which has six elements. Further, this action is faithful and 3-transitive, as is always the case for the action of the projective linear group on the projective line. This yields a map PGL(2, 5) → S_{6} as a transitive subgroup. Identifying PGL(2, 5) with S_{5} and the projective special linear group PSL(2, 5) with A_{5} yields the desired exotic maps S_{5} → S_{6} and A_{5} → A_{6}.^{[6]}
Following the same philosophy, one can realize the outer automorphism as the following two inequivalent actions of S_{6} on a set with six elements:^{[7]}
- the usual action as a permutation group;
- the six inequivalent structures as the projective line P^{1}(F_{5}) – the line has 6 points, and the projective linear group acts 3-transitively, so fixing 3 of the points, there are 3! = 6 different ways to arrange the remaining 3 points, which yields the desired alternative action.
Frobenius group
Another way: To construct an outer automorphism of S_{6}, we need to construct an "unusual" subgroup of index 6 in S_{6}, in other words one that is not one of the six obvious S_{5} subgroups fixing a point (which just correspond to inner automorphisms of S_{6}).
The Frobenius group of affine transformations of F_{5} (maps x ax + b where a ≠ 0) has order 20 = (5 − 1) · 5 and acts on the field with 5 elements, hence is a subgroup of S_{5}. (Indeed, it is the normalizer of a Sylow 5-group mentioned above, thought of as the order-5 group of translations of F_{5}.)
S_{5} acts transitively on the coset space, which is a set of 120/20 = 6 elements (or by conjugation, which yields the action above).
Other constructions
Ernst Witt found a copy of Aut(S_{6}) in the Mathieu group M_{12} (a subgroup T isomorphic to S_{6} and an element σ that normalizes T and acts by outer automorphism). Similarly to S_{6} acting on a set of 6 elements in 2 different ways (having an outer automorphism), M_{12} acts on a set of 12 elements in 2 different ways (has an outer automorphism), though since M_{12} is itself exceptional, one does not consider this outer automorphism to be exceptional itself.
The full automorphism group of A_{6} appears naturally as a maximal subgroup of the Mathieu group M_{12} in 2 ways, as either a subgroup fixing a division of the 12 points into a pair of 6-element sets, or as a subgroup fixing a subset of 2 points.
Another way to see that S_{6} has a nontrivial outer automorphism is to use the fact that A_{6} is isomorphic to PSL_{2}(9), whose automorphism group is the projective semilinear group PΓL_{2}(9), in which PSL_{2}(9) is index 4, yielding an outer automorphism group of order 4. The most visual way to see this automorphism is to give an interpretation via algebraic geometry over finite fields, as follows. Consider the action of S_{6} on affine 6-space over the field k with 3 elements. This action preserves several things: the hyperplane H on which the coordinates sum to 0, the line L in H where all coordinates coincide, and the quadratic form q given by the sum of the squares of all 6 coordinates. The restriction of q to H has defect line L, so there is an induced quadratic form Q on the 4-dimensional H/L that one checks is non-degenerate and non-split. The zero scheme of Q in H/L defines a smooth quadric surface X in the associated projective 3-space over k. Over an algebraic closure of k, X is a product of two projective lines, so by a descent argument X is the Weil restriction to k of the projective line over a quadratic étale algebra K. Since Q is not split over k, an auxiliary argument with special orthogonal groups over k forces K to be a field (rather than a product of two copies of k). The natural S_{6}-action on everything in sight defines a map from S_{6} to the k-automorphism group of X, which is the semi-direct product G of PGL_{2}(K) = PGL_{2}(9) against the Galois involution. This map carries the simple group A_{6} nontrivially into (hence onto) the subgroup PSL_{2}(9) of index 4 in the semi-direct product G, so S_{6} is thereby identified as an index-2 subgroup of G (namely, the unique such subgroup distinct from PGL_{2}(9) that also does not contain the Galois involution). Conjugation by any element of G outside of S_{6}defines the nontrivial outer automorphism of S_{6}.
Structure of outer automorphism
On cycles, it exchanges permutations of type (12) with (12)(34)(56) (class 2^{1} with class 2^{3}), and of type (123) with (145)(263) (class 3^{1} with class 3^{2}).
On A_{6}, it interchanges the 3-cycles (like (123)) with elements of class 3^{2} (like (123)(456)).
No other outer automorphisms
To see that none of the other symmetric groups have outer automorphisms, it is easiest to proceed in two steps:
- First, show that any automorphism that preserves the conjugacy class of transpositions is an inner automorphism. (This also shows that the outer automorphism of S_{6} is unique; see below.) Note that an automorphism must send each conjugacy class (characterized by the cyclic structure that its elements share) to a (possibly different) conjugacy class.
- Second, show that every automorphism (other than the above for S_{6}) stabilizes the class of transpositions.
The latter can be shown in two ways:
- For every symmetric group other than S_{6}, there is no other conjugacy class of elements of order 2 with the same number of elements as the class of transpositions.
- Or as follows:
Each permutation of order two (called an involution) is a product of k>0 disjoint transpositions, so it has cyclic structure 2^{k}1^{n-2k}. What's special about the class of transpositions (k=1)?
If one forms the product of two different transpositions τ_{1} and τ_{2}, then one always obtains either a 3-cycle or a permutation of type 2^{2}1^{n−4}, so the order of the produced element is either 2 or 3. On the other hand if one forms a product of two involutions σ_{1}, σ_{2}> that have type k>1, sometimes it happens that the product contains either
- two 2-cycles and a 3-cycle (for k=2 and n ≥ 7)
- a 7-cycle (for k=3 and n ≥ 7)
- two 4-cycles (for k=4 and n ≥ 8)
(for larger k, add to the permutations σ_{1}, σ_{2} of the last example redundant 2-cycles that cancel each other). Now one arrives at a contradiction, because if the class of transpositions is sent via the automorphism f to a class of involutions that has k>1, then there exist two transpositions τ_{1}, τ_{2} such that f(τ_{1}τ_{2})=f(τ_{1})f(τ_{2}) has order 6, 7 or 4, but we know that τ_{1}τ_{2} has order 2 or 3.
No other outer automorphisms of S_{6}
S_{6} has exactly one (class) of outer automorphisms: Out(S_{6} ) = C_{2}.
To see this, observe that there are only two conjugacy classes of S_{6} of order 15: the transpositions and those of class 2^{3}. Thus Aut(S_{6}) acts on these two conjugacy classes (and the outer automorphism above interchanges these conjugacy classes), and an index 2 subgroup stabilizes the transpositions. But an automorphism that stabilizes the transpositions is inner, so the inner automorphisms are an index 2 subgroup of Aut(S_{6}), so Out(S_{6}) = C_{2}.
More pithily: an automorphism that stabilizes transpositions is inner, and there are only two conjugacy classes of order 15 (transpositions and triple transpositions), hence the outer automorphism group is at most order 2.
Small n
Symmetric
For n = 2, S_{2} = C_{2} = Z/2 and the automorphism group is trivial (obviously, but more formally because Aut(Z/2) = GL(1, Z/2) = Z/2^{*} = 1). The inner automorphism group is thus also trivial (also because S_{2} is abelian).
Alternating
For n = 1 and 2, A_{1} = A_{2} = 1 is trivial, so the automorphism group is also trivial. For n = 3, A_{3} = C_{3} = Z/3 is abelian (and cyclic): the automorphism group is GL(1, Z/3^{*}) = C_{2}, and the inner automorphism group is trivial (because it is abelian).
Notes
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}
- ↑ Lam, T. Y., & Leep, D. B. (1993). "Combinatorial structure on the automorphism group of S_{6}". Expositiones Mathematicae, 11(4), 289–308.
- ↑ Lam, T. Y., & Leep, D. B. (1993). "Combinatorial structure on the automorphism group of S_{6}". Expositiones Mathematicae, 11(4), 289–308.
- ↑ Otto Hölder (1895), "Bildung zusammengesetzter Gruppen", Mathematische Annalen, 46, 321–422.
- ↑ ATLAS p. xvi
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}
References
- http://polyomino.f2s.com/david/haskell/outers6.html
- Some Thoughts on the Number 6, by John Baez: relates outer automorphism to icosahedron
- "12 points in PG(3, 5) with 95040 self-transformations" in "The Beauty of Geometry", by Coxeter: discusses outer automorphism on first 2 pages
- http://links.jstor.org/sici?sici=0002-9890(198206%2F07)89%3A6%3C407%3AOAO%3E2.0.CO%3B2-L
- http://links.jstor.org/sici?sici=0002-9890(199304)100%3A4%3C377%3ASOTCAO%3E2.0.CO%3B2-S
- http://links.jstor.org/sici?sici=0002-9890(196606%2F07)73%3A6%3C642%3ATOAO%3E2.0.CO%3B2-P
- http://links.jstor.org/sici?sici=0002-9890(195804)65%3A4%3C252%3AOATOH%3E2.0.CO%3B2-I