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 S₆, the symmetric group on 6 elements.

Summary

n	$\operatorname{Aut}(S_n)$	$\operatorname{Out}(S_n)$
$n\neq 2,6$	$S_n$	1
$n=2$	1	1
$n=6$	$S_6 \rtimes C_2$ ^[1]	$C_2$

n	$\operatorname{Aut}(A_n)$	$\operatorname{Out}(A_n)$
$n\geq 4, n\neq 6$	$S_n$	$C_2$
$n=1,2$	1	1
$n=3$	$C_2$	$C_2$
$n=6$	$S_6 \rtimes C_2$	$V=C_2 \times C_2$

Generic case

$n\neq 2,6$ : $\operatorname{Aut}(S_n) = S_n$ , and thus $\operatorname{Out}(S_n) = 1$ .

Formally,

S_n

is complete and the natural map

S_n \to \operatorname{Aut}(S_n)

is an isomorphism.

$n\neq 2,6$ : $\operatorname{Out}(A_n)=S_n/A_n=C_2$ , and the outer automorphism is conjugation by an odd permutation.
$n\neq 2,3,6$ : $\operatorname{Aut}(A_n)=\operatorname{Aut}(S_n)=S_n$

Formally, the natural maps

S_n \to \operatorname{Aut}(S_n) \to \operatorname{Aut}(A_n)

are isomorphisms.

Exceptional cases

$n=1,2$ : trivial:

\operatorname{Aut}(S_1)=\operatorname{Out}(S_1)=\operatorname{Aut}(A_1)=\operatorname{Out}(A_1)=1

\operatorname{Aut}(S_2)=\operatorname{Out}(S_2)=\operatorname{Aut}(A_2)=\operatorname{Out}(A_2)=1

$n=3$ : $\operatorname{Aut}(A_3)=\operatorname{Out}(A_3)=S_3/A_3=C_2$
$n=6$ : $\operatorname{Out}(S_6)=C_2$ , and $\operatorname{Aut}(S_6)=S_6 \rtimes C_2$ is a semidirect product.
$n=6$ : $\operatorname{Out}(A_6)=C_2 \times C_2$ , and $\operatorname{Aut}(A_6)=\operatorname{Aut}(S_6)=S_6 \rtimes C_2.$

The exceptional outer automorphism of S₆

Among symmetric groups, only S₆ has a (non-trivial) outer automorphism, which one can call exceptional (in analogy with exceptional Lie algebras) or exotic. In fact, Out(S₆) = C₂.^[2]

This was discovered by Otto Hölder in 1895.^[3]^[4]

This also yields another outer automorphism of A₆, 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 A₆, would be expected to have two outer automorphisms, not four. However, when A₆ 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 (Janusz & Rotman 1982).

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) S₅ → S₆
S₆ acts by conjugation on the six conjugates of this subgroup;

yielding a map S₆ → 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 S₆ (an inner automorphism)) yields an outer automorphism S₆ → S₆.

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 S₆ 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 S₅ → S₆

There is a subgroup (indeed, 6 conjugate subgroups) of S₆ which are abstractly isomorphic to S₅, and transitive as subgroups of S₆.

Sylow 5-subgroups

Janusz and Rotman construct it thus:

S₅ acts transitively by conjugation on its 6 Sylow 5-subgroups, yielding an embedding S₅ → S₆ 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, P¹(F₅), 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) → S₆ as a transitive subgroup. Identifying PGL(2, 5) with S₅ and the projective special linear group PSL(2, 5) with A₅ yields the desired exotic maps S₅ → S₆ and A₅ → A₆.^[6]

Following the same philosophy, one can realize the outer automorphism as the following two inequivalent actions of S₆ on a set with six elements:^[7]

the usual action as a permutation group;
the six inequivalent structures as the projective line P¹(F₅) – 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 S₆, we need to construct an "unusual" subgroup of index 6 in S₆, in other words one that is not one of the six obvious S₅ subgroups fixing a point (which just correspond to inner automorphisms of S₆).

The Frobenius group of affine transformations of F₅ (maps x $\mapsto$ ax + b where a ≠ 0) has order 20 = (5 − 1) · 5 and acts on the field with 5 elements, hence is a subgroup of S₅. (Indeed, it is the normalizer of a Sylow 5-group mentioned above, thought of as the order-5 group of translations of F₅.)

S₅ 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(S₆) in the Mathieu group M₁₂ (a subgroup T isomorphic to S₆ and an element σ that normalizes T and acts by outer automorphism). Similarly to S₆ acting on a set of 6 elements in 2 different ways (having an outer automorphism), M₁₂ acts on a set of 12 elements in 2 different ways (has an outer automorphism), though since M₁₂ is itself exceptional, one does not consider this outer automorphism to be exceptional itself.

The full automorphism group of A₆ appears naturally as a maximal subgroup of the Mathieu group M₁₂ 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 S₆ has a nontrivial outer automorphism is to use the fact that A₆ is isomorphic to PSL₂(9), whose automorphism group is the projective semilinear group PΓL₂(9), in which PSL₂(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 S₆ 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 S₆-action on everything in sight defines a map from S₆ to the k-automorphism group of X, which is the semi-direct product G of PGL₂(K) = PGL₂(9) against the Galois involution. This map carries the simple group A₆ nontrivially into (hence onto) the subgroup PSL₂(9) of index 4 in the semi-direct product G, so S₆ is thereby identified as an index-2 subgroup of G (namely, the unique such subgroup distinct from PGL₂(9) that also does not contain the Galois involution). Conjugation by any element of G outside of S₆defines the nontrivial outer automorphism of S₆.

Structure of outer automorphism

On cycles, it exchanges permutations of type (12) with (12)(34)(56) (class 2¹ with class 2³), and of type (123) with (145)(263) (class 3¹ with class 3²).

On A₆, it interchanges the 3-cycles (like (123)) with elements of class 3² (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 S₆ 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 S₆) stabilizes the class of transpositions.

The latter can be shown in two ways:

For every symmetric group other than S₆, 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^k1^n-2k. What's special about the class of transpositions (k=1)?

If one forms the product of two different transpositions τ₁ and τ₂, then one always obtains either a 3-cycle or a permutation of type 2²1ⁿ⁻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 σ₁, σ₂> 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 σ₁, σ₂ 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 τ₁, τ₂ such that f(τ₁τ₂)=f(τ₁)f(τ₂) has order 6, 7 or 4, but we know that τ₁τ₂ has order 2 or 3.

No other outer automorphisms of S₆

S₆ has exactly one (class) of outer automorphisms: Out(S₆ ) = C₂.

To see this, observe that there are only two conjugacy classes of S₆ of order 15: the transpositions and those of class 2³. Thus Aut(S₆) 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(S₆), so Out(S₆) = C₂.

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, S₂ = C₂ = 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 S₂ is abelian).

Alternating

For n = 1 and 2, A₁ = A₂ = 1 is trivial, so the automorphism group is also trivial. For n = 3, A₃ = C₃ = Z/3 is abelian (and cyclic): the automorphism group is GL(1, Z/3^*) = C₂, and the inner automorphism group is trivial (because it is abelian).

Notes

↑ Janusz, Gerald; Rotman, Joseph (June–July 1982), "Outer Automorphisms of S₆", The American Mathematical Monthly 89 (6): 407–410, JSTOR 2321657
↑ Lam, T. Y., & Leep, D. B. (1993). "Combinatorial structure on the automorphism group of S₆". Expositiones Mathematicae, 11(4), 289–308.
↑ Lam, T. Y., & Leep, D. B. (1993). "Combinatorial structure on the automorphism group of S₆". Expositiones Mathematicae, 11(4), 289–308.
↑ Otto Hölder (1895), "Bildung zusammengesetzter Gruppen", Mathematische Annalen, 46, 321–422.
↑ ATLAS p. xvi
↑ Carnahan, Scott (2007-10-27), "Small finite sets", Secret Blogging Seminar, notes on a talk by Jean-Pierre Serre.
↑ Snyder, Noah (2007-10-28), "The Outer Automorphism of S₆", Secret Blogging Seminar

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