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Notes and Exercises on G–sets
May 14, 2010
These notes collect some useful definitions and results pertaining to G-sets. They supplement
Professor Lampe’s excellent set of notes on the subject [1]. In the second section below, I (attempt
to) solve some of the exercises appearing in Dr. Lampe’s notes. (The solutions have not been
checked for correctness.) The notation is as described in [1]; we repeat a few of the definitions, but
most of the notation set down in [1] is taken for granted.
1
Definitions
Let G be a group, A = hA; Gi a G-set, and let Sym(A) denote the group of permutations of A.
orbits For a ∈ A, the one-generated subalgebra [a] ∈ Sub[A] is called the orbit of a in A. It is
easily verified (see exercise 1 of section 2) that [a] is equal to the set Ga := {ga | g ∈ G}, and
we often use the more suggestive Ga when refering to this orbit.
The orbits of the G-set A partition the set A into disjoint equivalence classes. The equivalence
relation ∼ is defined as follows: for (x, y) ∈ A × A,
x∼y
⇔
gx = y
for some g ∈ G.
In fact, ∼ is a congruence of the algebra A since, for all (x, y) ∈ A × A, and for all g ∈ G,
x∼y
⇒
gx ∼ gy.
Thus, as mentioned above, each orbit is indeed a subalgebra of A.
It can be helpful to keep in mind that A is the disjoint union of the orbits. That is, if
{a1 , . . . , ar } is a full set of ∼-class representatives, then A = ∪· ri=1 Gai .
transitive A G-set with only one orbit is called transitive. Equivalently, hA, Gi is a transitive G-set
iff (∀a, b ∈ A)(∃g ∈ G)(ga = b). In this case, we say that G acts transitively on A, and
occasionally we refer to the group G itself as a transitive group.
stabilizer For a ∈ A, the set StabG (a) := {g ∈ G | ga = a} is called the stabilizer of a. It is easy to
verify (see exercise below) that StabG (a) is a subgroup of G. An alternative notation for the
stabilizer, which we adopt below, is Ga := StabG (a).
faithful Let λ : G → G 6 Sym(A) denote the permutation representation of G; i.e., λ(g) = g (g ∈ G).
Then
ker λ = {g ∈ G | ga = a for all a ∈ A} =
\
a∈A
1
StabG (a) =
\
a∈A
Ga .
(1)
Notes and Exercises on G–sets
May 14, 2010
By Noether isomorphism, G/ ker λ ∼
= λ[G] 6 Sym(A). We say that the representation λ of G
is faithful, or that G acts faithfully on A, just in case ker λ = (e). In this case λ embeds G
in the group Sym(A), we refer to G as a permutation group acting on A, and we call hA, Gi
a permutational algebra.
core If H 6 G are groups, the core of H in G, denoted coreG (H), is the largest normal subgroup
of G that is contained in H. It is easy to see that
coreG (H) =
\
gHg −1 .
g∈G
A subgroup H is called core-free provided coreG (H) = (e), the identity in G.
As we will see below (Proposition 2), elements in the same orbit of a G-set have conjugate
stabilizers. Specifically, if a, b ∈ A and g ∈ G are such that ga = b, then
Gb = Gga = g Ga g −1 .
(2)
If the G-set happens to be transitive, then it is faithful iff the stabilizer Ga is core-free in G,
for any (hence every) a ∈ A. To see this, note that if G acts transitively, then Ga = A and,
(1) and (2) imply
ker λ =
\
a∈A
Ga =
\
g∈G
Gga =
\
g Ga g −1 .
g∈G
Thus Ga is core-free iff ker λ = (e) iff G acts faithfully on A.
regular In case G is a transitive permutation group, we say that G is regular, or that G acts regularly
on the set A, or that λ : G → G is a regular representation, provided Ga = (e) for each a ∈ A;
equivalently, only the identity fixes any point.1
blocks Let A = hA, Gi be a transitive G-set. A nonempty subset B ⊆ A is called a block for A if
for each g ∈ G either gB = B or gB ∩ B = ∅. (We will call such B a block if the underlying
G-set is clear from context.)
For example, every transitive G-set hA, Gi has A and the singletons {a} (a ∈ A) as blocks;
these are called the trivial blocks. Any other block is called nontrivial. A block which is
minimal in the set of all blocks of size > 1 is called a minimal block.
If B and C are blocks of a transitive G-set containing a common point, then B ∩ C is also
a block. More, generally, any intersection of blocks containing a common point is again a
1
It seems some geometers like to call the action of a regular permutation group a “free action.”
2
Notes and Exercises on G–sets
May 14, 2010
block.
The importance of blocks arises from the following observation. Suppose that A = hA, Gi is a
transitive G-set and that B is a block for A. Put β = {gB | g ∈ G}. Then the sets in β form
a partition of A and each element of β is a block. We call β the system of blocks containing
B. Now G acts on β in an obvious way, and this new action may give useful information
about G provided B is not a trivial block.
primitive Let A = hA, Gi be a transitive G-set. We say that the group G is primitive if A has
no nontrivial blocks; otherwise G is called imprimitive. Note that we only use the terms
“primitive” and “imprimitive” with reference to a transitive G-set.
1. Theorem: Let A = hA, Gi be a transitive G-set and let a ∈ A. Let B be the set of all
blocks B with a ∈ B. Let [Ga , G] ⊆ Sub[G] denote the set of all subgroups of G containing
Ga . Then there is a bijection Ψ : B → [Ga , G] given by Ψ(B) = G(B), with inverse mapping
Φ : [Ga , G] → B given by Φ(H) = Ha = {ha | h ∈ H}. The mapping Ψ is order-preserving
in the sense that if B1 , B2 ∈ B then B1 ⊆ B2 ⇔ Ψ(B1 ) 6 Ψ(B2 ).
Briefly, the poset hB, ⊆i is order-isomorphic to the poset h[Ga , G], 6i.
Corollary: Let G act transitively on a set with at least two points. Then G is primitive if
and only if each stabilizer Ga is a maximal subgroup of G.
Since the point stabilizers of a transitive group are all conjugate, one stabilizer is maximal
only when all of the stabilizers are maximal. In particular, a regular permutation group is
primitive if and only if it has prime degree.
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Notes and Exercises on G–sets
1.1
May 14, 2010
Interlude: a Galois correspondence
This section contains a few basic, anciliary results, some of which I’ve seen elsewhere, and some of
which I derived myself.2 Most likely everything in this section is well-known, as it’s all quite basic.
I. We extend the action of the group G on the set A to subsets of A as follows:3
Let B ⊆ A and S ⊆ G be arbitrary non-empty subsets.
For each g ∈ G, let gB := {gb | b ∈ B} ⊆ A, and define
G(B) := {g ∈ G | gB = B}
InvG (B) := {g ∈ G | gb = b for all b ∈ B} =
\
Gb
b∈B
FixA (S) := {a ∈ A | ga = a for all g ∈ S} = {a ∈ A | S ⊆ Ga }.
It is not hard to see that G(B) is a subgroup of G. In fact, G(B) is the largest subgroup which
acts on B. Thus, B is a subuniverse of A provided G(B) = G. The set InvG (B) is called
the pointwise stabilizer of B, while G(B) is sometimes called the global stabilizer or setwise
stabilizer of B.
The following are some easily checked, useful consequences of the definitions:
(i) Let λ : G → Sym(A) denote the permutation representation of G on A. For each g ∈ G,
the corresponding action (appearing in the definitions above) is g = λ(g). Then,
ker λ =
\
a∈A
Ga = InvG (A) P G
and G/InvG (A) ∼
= G 6 Sym(A). Of course, if G happens to be simple (i.e., the only
normal subgroups of G are (e) and G), then G ,→ Sym(A). Also, if G acts transitively but
not faithfully, then the group G/InvG (A) acts faithfully and transitively (i.e., G/InvG (A)
is a transitive permutation group).
(ii) Let λ̂ denote the restriction of λ to G(B), and view the image λ̂[G(B)] as a group of
actions on the set B (i.e., restrict both the domain and range of λ). The result is a
2
The exploration of the Galois correspondence given by Fix and Inv was inspired by Professor Lampe’s elegant,
abstract treatment of Galois theory [2]. Unfortunately, group theorists seem to have already assigned the name Fix
to the operation on subgroups, so the use of Inv and Fix here might seem to be the reverse of how they appear in
Professor Lampe’s notes.
3
cf. J. Rose, A Course on Group Theory (Cambridge: 1978), p. 71.
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Notes and Exercises on G–sets
May 14, 2010
permutation representation of G(B) on B:
λ̂ : G(B) → Sym(B),
with ker λ̂ = InvG(B) (B) = InvG (B) P G(B), and so G(B)/InvG (B) is isomorphic to
a subgroup of Sym(B). (In particular, |G(B)/InvG (B)| divides |Sym(B)|, a fact which
occasionally comes in handy.)
To illustrate a special case of the foregoing, suppose B = Gai is an orbit (or any subuniverse of hA, Gi for that matter), then G acts on B, and the corresponding permutation
representation of G on B has kernel InvG (B) P G, and image a subgroup of Sym(B).
Therefore, G/InvG (B) is isomorphic to a subgroup of Sym(B).
(iii) G(gB) = g −1 G(B)g, for all g ∈ G.
(iv) InvG (gB) = g −1 InvG (B)g, for all g ∈ G.
(v) gFixA (S) = FixA (gSg −1 ), for all g ∈ G.
(vi) FixA maps normal subgroups of G to subuniverses of A.
(vii) InvG maps subuniverses of A to normal subgroups of G.
II. One conclusion we can draw from the facts in part I. is the following general Cayley theorem:4
Proposition: Suppose that the finite group G acts faithfully on the finite set A.
Let
Ga1 , . . . , Gar be a complete set of orbits, with respective cardinalities |Gai | = ni . Then
the group G can be embedded in the group5 Σn1 × · · · × Σnr .
Proof: As noted in I, G defines an action on each orbit. Let λi denote the range-restricted
permutation representation of G as an action on the orbit Gai . Then, λi : G → Sym(Gai ),
and
ker λi = InvG (Gai ) =
\
g∈G
Ggai =
\
g Gai g −1 .
g∈G
Moreover,
G/ ker λi ∼
= λi (G) 6 Sym(Gai ) ∼
= Σni .
T
As G acts faithfully on A, (e) = ker λ = ri=1 ker λi . Therefore, the map g 7→ (g ker λ1 , . . . , g ker λr )
is a well-defined monomorphism of G into G/ ker λ1 × · · · × G/ ker λr ∼
= Σn × · · · × Σn .
1
4
5
ibid., ex. 188.
Σn denotes the symmetric group on n letters.
5
r
Notes and Exercises on G–sets
2
May 14, 2010
Exercises and Solutions
Suppose G is a group and A = hA; Gi is a G–set.
(1) If [a] denotes the subalgebra generated by a ∈ A, then [a] = {g(a) | g ∈ G}.
Proof: Recall, if A = hA; Oi is an algebra, a subset B ⊆ A is called a subuniverse of
A provided B is closed under the operations O. For any subset X ⊆ A, the subalgebra
generated by X is defined by
[X] :=
\
{B ⊆ A | B is a subuniverse of A and X ⊆ B}.
Suppose A = hA; Gi is a G–set and a ∈ A. Denote the subset {ga | g ∈ G} ⊆ A by Ga. We
show [a] = Ga.
If e = idG , then e(a) = a ∈ Ga. Also, Ga is a subuniverse of A since it’s closed under the
operations G. Therefore, [a] ⊆ Ga. On the other hand, if B is any subuniverse of A that
contains a, then Ga ⊆ B, since B is closed under G. In other words, Ga is contained in every
subuniverse of A that contains a. Thus, Ga ⊆ [a].
(2) If a, b ∈ A and b ∈ [a], then [b] = [a].
Proof: First note that b ∈ [a] implies [b] ⊆ [a] since [a] is a subuniverse of A containing b.
Also, b ∈ [a] implies b = ga for some g ∈ G, by (1). Therefore, since g −1 · g = idA ,
b = ga
⇔
g −1 (b) = a
⇒
a ∈ [b].
The last implication comes from (1) again. Finally, a ∈ [b] implies [a] ⊆ [b], as in the first
sentence of the proof.
(3) The one-generated subalgebras of A form a partition of A.
Proof: The relation defined by a ∼ b ⇔ [a] = [b] is obviously an equivalence relation and the
equivalence classes are the one-generated subalgebras.
(4) Suppose A = B ∪ C and B and C are distinct one-generated subalgebras of A. (We let B =
hB; ( g|B : g ∈ G)i and similarly for C.) If Φ ∈ Con(B) and Ψ ∈ Con(C), then Φ∪Ψ ∈ Con(A).
(5) State and prove a generalization of (4).
(6) Proposition 1: The stabilizer Ga is a subgroup of G.
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Notes and Exercises on G–sets
May 14, 2010
Proof: We show that if g, h ∈ Ga , then h−1 ∈ Ga and gh−1 ∈ Ga .
h ∈ Ga ⇔ a = h(a) ⇔ h−1 (a) = h−1 · h(a) = a ⇔ h−1 ∈ Ga
and gh−1 (a) = g · h−1 (a) = ga = a, since g, h−1 ∈ Ga .
(7) Proposition 2: Elements in the same orbit have conjugate stabilizers. That is, if a ∈ A and
b = ga, then Gb = g Ga g −1 .
Proof: If b = ga, then
Gb = Gga = {h ∈ G | h(ga) = ga} = {h ∈ G | g −1 hg(a) = a}.
Letting f = g −1 hg in the last expression, we see that
Gb = {gf g −1 ∈ G | f (a) = a} = g Ga g −1 .
(8) Theorem 1: If A = hA; Gi is a transitive G-set, then A is isomorphic to the G-set
G/Ga := hG/Ga ; {λ̂g : g ∈ G}i
for any a ∈ A.
Proof: Suppose A = hA; Gi is a transitive G-set, so A = {ga | g ∈ G} for any a ∈ A. Recall
(cf. example (4) of [1]), the G-set G/Ga has group action λ̂g defined, for each g ∈ G and each
xGa ∈ G/Ga , by λ̂g (xGa ) = gxGa .
Let GΛ denote the G-set hG; {λg : g ∈ G}i – that is, G acting on itself by left multiplication.
Fix a ∈ A, and define ϕa : G → A by ϕa (x) = x(a) for each x ∈ G. Then ϕa is a homomorphism
from GΛ into A – that is, ϕa respects operations:6
ϕa (λg (x)) = ϕa (gx) = gx(a) = g · x(a) = gϕa (x).
Moreover, since A is transitive, ϕa (G) = {ga | g ∈ G} = A, so ϕa is an epimorphism. Therefore,
GΛ / ker ϕa ∼
= A. To complete the proof, we check that GΛ / ker ϕa = G/Ga .
6
In general, if A = hA, F i and B = hB, F i are two algebras of the same similarity type, then ϕ : A → B is a
homomorphism provided
ϕ(f A (a1 , . . . , an )) = f B (ϕ(a1 ), . . . , ϕ(an ))
whenever f A is an n-ary operation of A, f B is the corresponding n-ary operation of B, and a1 , . . . , an are arbitrary
elements of A. (Note that a one-to-one correspondence between the operations of two algebras of the same similarity
type is assumed, and required for the definition of homomorphism to make sense.)
7
Notes and Exercises on G–sets
May 14, 2010
Recall that ker ϕa is the equivalence relation on G given by ker ϕa = {(x, y) ∈ G × G | ϕa (x) =
ϕa (y)}. If, for each x ∈ G, we denote the equivalence class containing x by x/ ker ϕa = {y ∈
G | (x, y) ∈ ker ϕa }, then the universe of GΛ / ker ϕa is G/ ker ϕa = {x/ ker ϕa | x ∈ G}. Note
that, for each x ∈ G,
x/ ker ϕa = {y ∈ G | (x, y) ∈ ker ϕa } = {y ∈ G | ϕa (x) = ϕa (y)}
= {y ∈ G | x(a) = y(a)} = {y ∈ G | idA (a) = x−1 y(a)}
= {y ∈ G | x−1 y ∈ Ga } = {y ∈ G | xGa = yGa }
= xGa .
These are precisely the elements of the universe G/Ga . That is, the universes of GΛ / ker ϕa
and G/Ga are the same. As the operations (left multiplication by g ∈ G) are also the same,
the algebras are identical.
Remarks: Leaving out a few of the details given above, Pálfy explains this situation succinctly
as follows:
“If G acts transitively on A, then choosing an element a ∈ A, the elements of A are in one-to-one
correspondence with the right7 cosets of the stabilizer Ga , namely, b ↔ {g ∈ G | ag = b}. Thus
hA; Gi ∼
= hG/Ga ; Gi. So there is a one-to-one correspondence between the transitive actions of
G and the conjugacy classes of subgroups in G.” [italics added]
Apparently, it is immediate (to Pálfy, at least) that the one-to-one correspondence between
elements of A and cosets of the stabilizer respects operations (as required by the conclusion
that this correspondence is an isomorphism: hA; Gi ∼
= hG/Ga ; Gi.)
Pálfy also gives the following proof sketch of Theorem 2 of the G-sets notes (we prove it below):
“If ϕ : hA; Gi → hB; Gi is a homomorphism, then clearly Ga 6 Gϕ(a) . Conversely, if H 6 K 6
G, then Hx → Kx gives a well-defined homomorphism hG/H; Gi → hG/K; Gi. Thus, if G acts
transitively on A, then Con(A; G) ∼
= [Ga , G].”
References
[1] William
A.
Lampe.
Math
611
notes:
Groups
acting
on
sets
or
G–sets.
http://www.math.hawaii.edu/~williamdemeo/611_07_G-sets.pdf, May 2009.
7
Clearly it makes no difference whether we use right cosets throughout, instead of left cosets, as long as we’re
consistent.
8
Notes and Exercises on G–sets
[2] William
A.
Lampe.
May 14, 2010
Math
612
notes:
Galois
http://www.math.hawaii.edu/~williamdemeo/612_08_Galois.pdf, May 2009.
9
theory.