The previous two sections exhibit two families of finite simple groups, namely the alternating groups
An for n ≥ 5and the groups PSL2(q) = PSL2(F)where F is a finite field with |F| = q ≥ 4. A
small number of groups are common to both families via isomorphisms whose subtly ranges from the non-obvious to the miraculous. We list these exceptional isomorphisms, and a few others.
S3 'GL2(2) = PSL2(2) A4 'PSL2(3) A5 'PSL2(4) 'PSL2(5) PSL2(7) 'GL3(2) = PSL3(2) A6 'PSL2(9) A8 'PSL4(2). (9)
The first two, as well as the isomorphismA5 'PSL2(4), arise easily from the theory of group actions
in the next section. For the remaining exceptional isomorphisms, see sections5.4.1, 5.4.2, 8.5.2and
??.
4
Group actions
We say that groupGactson the setX if there is a homomorphismϕ:G→SX fromGinto the group
permutationϕgofX, which sends anyx∈Xto an elementϕ(g)x. Ifϕis understood or is completely
general, we usually omit it from the notation, writinggxorg·xinstead ofϕ(g)x.
If(X, ϕ) and(Y, ψ) are two G-sets, a function f : X → Y is called G-equivariant if f(ϕ(g)x) =
ψ(g)f(x)for allg ∈Gandx∈X. We say that(X, ϕ)and(Y, ψ)areequivalentG-setsif there exists aG-equivariant bijectionf :X →Y.
The notion of a G-set generalizes the notion of a group. For we can regard the action as a map
G×X →X, given by(g, x)7→g·xsuch thatg·(g0·x) = (gg0)·xfor allg, g0 ∈Gandx∈X. Some standard terminology associated with group actions is as follows.
Thestabilizerorfixerof a pointx∈X is the subgroup ofGgiven by
Gx ={g ∈G: g·x=x} ≤G.
Theorbitof an elementx∈X is the subset ofX given by
G·x={g·x: g ∈G} ⊆X.
Orbits are equivalence classes under the equivalence relationx∼yify=g·xfor someg ∈G. Hence two orbits are either equal or disjoint; the orbits form a partition ofX.
Thekernel of a group action ϕ : G → SX is which is the normal subgroup of Gconsisting of the
element acting trivially onX. We have
kerϕ= \
x∈X
Gx,
AG-action onXisfaithfulifkerϕis trivial. Equivalently, the action is faithful if no nontrivial element ofGacts trivially onX. In this case,Gis isomorphic to a subgroup ofSX.
Finally, a group action isfreeif g·x = xfor somex ∈ X impliesg = 1. That is, a group action is free iff all stabilizers are trivial. Clearly free actions are faithful. An example of a free action is where a subgroupH of a groupGacts onGby left multiplication: h·x =hx. Here,Gis the set andH is the group which is acting. The orbits are the right cosetsHx.
Proposition 4.1 For allg ∈Gandx∈X, we have
Gg·x =gGxg−1.
In particular, the stabilizers of all elements of the same orbit are conjugate.
AG-action onX istransitiveif for allx, y ∈Xthere existsg ∈Gsuch thatg ·x=y. Equivalently, the action is transitive iff X consists of a single G-orbit. For a general group action, each orbit is a transitiveG-set. Thus, transitive group actions are the essential ones. An example of a transitive group action is whereX =G/H, for some subgroupH ≤G, and the action isg·xH =gxH. We will see
that all transitiveG-actions are of this form. More generally, a G-action onX isk-transitive ifGis transitive onk-element subsets ofX. This is a measure of the strength of transitivity.
The Main Theorem of Group Actions
If a groupGacts on a setX, then for eachx∈Xwe have aG-equivariant bijection
f :G/Gx
∼
−→G·x, given by f(gGx) =g·x. In particular, any transitive group action is equivalent to an action on cosets.
Proof: The map f is well-defined because for allh ∈ Gx we have (gh)·x = g ·(h·x) = g ·x.
The mapf is injective because ifg·x =g0 ·xtheng−1g0 ·x = x, sog−1g0 ∈ Gx, which means that
gGx=g0Gx. The mapf is surjective, by the definition of the orbitG·x. Finally, for allg, g0 ∈Gand
x∈Xwe have
f(g·g0Gx) =f(gg0Gx) = (gg0)·x=g·(g0 ·x) =g·f(g0Gx),
which shows thatf isG-equivariant.
As a corollary, we have one of the most useful formulas in group theory.
The Counting Formula. LetG be a finite group acting on a setX. Then the cardinality of an orbit equals the index of the stabilizer of any point in the orbit. That is, for anyx∈X we have
|G| |Gx|
=|G·x|. (10)
Note that the right hand side of this equation depends only on the orbit, while the left side appears to depend on the stabilizer of a particular point in the orbit. In fact, the left side depends only on the orbit of the stabilizer. By Prop. 4.1all stabilizers in a given orbit are conjugate, hence have the same order. LetO1, . . . , Ok be the orbits ofGinX, and choosexi ∈ Oi. Applying the counting formula to each
orbit, we have the weaker but still useful formula
|X|= k X i=1 |Oi|= k X i=1 [G:Gxi]. (11)
4.1
The left regular action
Any group Gacts on itself by left multiplication. More precisely, this action is given by the homo- morphismL : G → SG such that Lgx = gxfor all g, x ∈ G. This is called the left regular action;
one easily checks that it is free and transitive. Moreover, any free transitive G-action on a set X is isomorphic to the left regular action (Exercise ...). If Gis finite, say |G| = n, thenSG ' Sn via a
labelling of the elements ofG. Since the left regular action is faithful, this proves:
Proposition 4.2 A finite group Gof order n is isomorphic to a subgroup of Sn, via the left regular actionL:G→Sn.
In other words,Sncontains every group of ordernas a subgroup.
What are the cycle types of the elements of L(G)? For any permutation σ ∈ Sn, the numbers in
the cycle typeσ are the sizes of the orbits of hσi on {1,2, . . . , n}. Since Gacts freely on itself, the subgrouphgialso acts freely onGand the orbits ofhgionGare just the cosets ofhgi. The orderdof
gdividesnand there aren/dcosets ofhgiinG, all of sized. This proves
Proposition 4.3 LetGbe a finite group of ordern, letg ∈G, and letdbe the order ofg. Then under the left regular actionL:G→Snthe cycle type ofLg is a product ofn/dcycles of lengthd.
This has the following surprising corollary.
Corollary 4.4 SupposeGis a group of orderncontaining an element of orderdwheredis even and
n/dis odd. ThenGhas a normal subgroup of index two. In particular,Gcannot be simple.
Proof: Ifg ∈Ghas even orderdwithn/dodd, thenLg is a product of an odd number of even cycles,
sosgn(Lg) = −1. Hence the homomorphismsgn◦L : G → {±1}is nontrivial, soker(sgn◦L) is a
normal subgroup ofGof index two.
From this, we can prove another partial converse to Lagrange’s theorem.
Corollary 4.5 SupposeGis a group of order2mwheremis odd. ThenGhas a normal subgroup of orderm.
Proof: By Prop.2.2there existsg ∈Gof order two, satisfying the conditions of Cor. 4.4.
Since a groupGalso acts on itself by right multiplication, we also have theright regular actiongiven by the homomorphismR : G → SG given by ρgx = xg−1. [Check that this is a group action- in so
doing, you’ll see why we need the inverse.] Analogues of the above results all hold forRas well.