Alphabeta MathGroup Actions, Orbits, Stabilisers and Cayley's Theorem
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1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
2 · Logical flowchart
3 · Items
Definition
Left group actions, transitive actions, and faithful actions
Theorem
Actions of G on X correspond exactly to homomorphisms G toSym(X)
Definition
The orbit G· x and stabilizer Gₓ of a point in a group action
Lemma
The stabilizer Gₓ is a subgroup of G
Theorem
The orbits of a group action are the equivalence classes of x∼ y iff y=g· x for some g, and hence partition the acted-on set
Theorem
Cayley's theorem: every group G is isomorphic to a subgroup of Sym(G)
Definition
Equivariant maps and isomorphisms of group actions
Definition
A free group action has no nonidentity element fixing a point
Definition
The fixed-point sets Xᵍ and X^G of a group action
Theorem
Orbit-stabiliser: G/Gₓ→ G· x, gGₓ↦ g· x, is a well-defined bijection
Corollary
Orbit-stabiliser cardinality: |G· x|=[G:Gₓ] whenever either side is finite, and |G|=|Gₓ| |G· x| for finite G
Lemma
If y=g· x, then G_y=gGₓg⁻¹
Definition
The core Core_G(H)=⋂_g∈ GgHg⁻¹ of a subgroup
Lemma
Core_G(H) is the largest normal subgroup of G contained in H
Theorem
Left multiplication on G/H is transitive, has stabiliser H at H, and has kernel Core_G(H)
Theorem
If [G:H]=n<∞, then Core_G(H) is normal in G, [G:Core_G(H)]∣ n!, and only finitely many subgroups contain H
Theorem
Every transitive G-set is equivariantly isomorphic to G/Gₓ for any chosen point x
Corollary
A transitive action is faithful exactly when a point stabiliser is core-free
Definition
The conjugacy class Cl_G(x) and centralizer C_G(x) of an element
Definition
The normalizer N_G(H)={g∈ G:gHg⁻¹=H} of a subgroup
Lemma
C_G(x) and N_G(H) are subgroups of G
Theorem
G/C_G(x)toCl_G(x) is a bijection, so |Cl_G(x)|=[G:C_G(x)] whenever these cardinalities are finite
Theorem
The conjugates of H are in bijection with G/N_G(H) and, for finite G, number [G:N_G(H)]
Theorem
The conjugates of a proper subgroup do not cover a finite group
Theorem
The class equation |G|=|Z(G)|+∑ᵢ [G:C_G(xᵢ)] for a finite group
Definition
A finite p-group has order pⁿ for a prime p and some n∈ℕ
Lemma
Every subgroup of a finite p-group has order a power of p
Corollary
Every subgroup of index p in a finite p-group is normal
Theorem
If a finite p-group P acts on a finite set X, then |X|≡|X^P| (mod p)
Corollary
A finite p-group action on X has a global fixed point whenever p∤|X|
Theorem
Every nontrivial finite p-group has nontrivial center, in fact p divides |Z(P)|
Lemma
If G/Z(G) is cyclic, then G is abelian
Corollary
Every group of order p², for prime p, is abelian
Theorem
Every nontrivial normal subgroup of a finite p-group meets the center nontrivially
Theorem
Cauchy's theorem: if a prime p divides |G|, then G has an element of order p
Theorem
Cauchy-Frobenius orbit counting: |G| |X/G|=∑_g∈ G|Xᵍ| for a finite group action
Theorem
Jordan's derangement theorem: every transitive action of a finite group on a finite set with more than one element has a nonidentity element with no fixed points
Sources
- Brosnan, Group actions
- Brosnan, Orbits and stabilizers
- Brosnan, Cayley's theorem
- P. Brosnan, Undergraduate Algebra Notes, 3.14: G-Sets
- T. W. Judson, Abstract Algebra: Theory and Applications, 14.1
- T. W. Judson, Abstract Algebra: Theory and Applications, 14.3
- P. Brosnan, Undergraduate Algebra Notes, 3.14: G-Sets, Theorem 3.107
- T. W. Judson, Abstract Algebra: Theory and Applications, 14.1, Theorem 14.11
- K. Conrad, Group Actions, Theorem 6.8
- P. Brosnan, Undergraduate Algebra Notes, 3.14: G-Sets, Proposition 3.102 and Corollary 3.104
- P. Brosnan, Undergraduate Algebra Notes, 3.14: G-Sets, Lemma 3.105 and Theorem 3.107
- T. W. Judson, Abstract Algebra: Theory and Applications, 14.2
- P. Brosnan, Undergraduate Algebra Notes, 3.14: G-Sets, Corollary 3.109
- P. Brosnan, Undergraduate Algebra Notes, 3.14: G-Sets, Corollary 3.110
- K. Conrad, Group Actions, Theorem 6.10
- T. W. Judson, Abstract Algebra: Theory and Applications, 14.2, The Class Equation
- K. Conrad, Group Actions, Section 4
- K. Conrad, Group Actions, Corollary 6.4
- K. Conrad, Group Actions, Theorem 4.1
- K. Conrad, Group Actions, Corollary 4.2
- K. Conrad, Group Actions, Theorem 5.1
- T. W. Judson, Abstract Algebra: Theory and Applications, Corollary 14.16, proof
- K. Conrad, Group Actions, Corollary 5.2
- K. Conrad, Group Actions, Theorem 5.3
- K. Conrad, Group Actions, Theorem 5.4
- T. W. Judson, Abstract Algebra: Theory and Applications, 14.3, Burnside's Counting Theorem
- K. Conrad, Group Actions, Theorem 3.29
- K. Conrad, Group Actions, Theorem 6.6