Fichas
Permutation Groups
Fichas del conjunto (30)
- 1GroupA group G is a non-empty set G, together with a binary operation called multiplication s.t. the following hold:
(1) for all g,h∈G, their product gh∈G
(2) ∃1∈G s.t. 1g=g1=g for all g∈G
(3) ∀g∈G ∃g^(-1)∈G s.t. gg^(-1)=g^(-1)g=1
(4) The group multiplication is associative: g(hk)=(gh)k ∀g,h,k∈G - 2Symmetric GroupS(n) is the symmetric group of all permutations on the set {1,...,n}
- 3Alternating GroupA(n) is the group of all even permutations, that is those which are the product of an even number of transpositions (i,j) (a subgroup of S(n))
- 4ConjugationConjugation in a group is denoted by h^(g)=g^(-1)hg
- 5CosetIf H≤G, then a (right) coset of H is a set H(g)= {hg:h∈H}. These are all equal or disjoint and partition G.
- 6Lagrange's TheoremLet G be a group and H≤G a subgroup. Then |G|=|G:H||H| where |G:H| is the number of cosets of H in G, called the index of H in G
- 7ConjugateWe say two elts h and k conjugate is there exists g∈G s.y. hᵍ=k. Likewise two subgroups H,K≤G are conjugate if there exists g∈G s.t Hᵍ=K. Being conjugate is an equivalence relation
- 8Normal SubgroupWritten N◅G is a subgroup N≤G s.t ∀h∈N and g∈G hᶢ∈N
- 9Quotient GroupWhen N is a normal subgroup of G, we may form the quotient group G/N whose elements are cosets of N and multiplication is given by Ng·Nh=Ngh
- 10HomomorphismA (group) homomorphism is a well-defined map φ:G→H between two groups G and H which preserves the multiplicative structure. In other words, φ(gk)=φ(g)φ(k) for all g,k∈G
- 11IsomorphismA bijective homomorphism is called an isomorphism. Write G≅H when G and H are isomorphic.
- 121st Isomorphism TheoremLet G and H be groups and φ:G→H be a homomorphism. Then N:=kerφ is a normal subgroup of G and the induced map
-φ-:G/N→Im(φ)≤H
Ng↦φ(g)
is an isomoorphism between the quotient group G/N and the image Im(φ) - 13ActionsLet Ω be a non-empty finite set and G a group. We say that G acts on Ω if there is a map φ : Ω × G → Ω s.t
(1) µ(µ(α, g), h) = µ(α, gh)
(2) µ(α, 1) = α
for all α ∈ Ω and g, h ∈ G. If Ω is finite, then we say the action, or G, has degree |Ω|. - 14Kernel and FaithfullyThe kernel of the action is
{g ∈ G : αg = α for all α ∈ Ω} ◅ G
We say that G acts faithfully if it acts with no non-trivial kernel - 15OrbitLet G be a group acting on a set Ω. We define the orbit containing α to be
αG :={β ∈ Ω : ∃g ∈ G s.t. β = αg}
={αg : g ∈ G} - 16StabiliserDefine the stabiliser of α in G to be
Gα := {g ∈ G : αg = α} - 17Transitive and IntransitiveA group G acting on a set Ω is transitive if for all α, β ∈ Ω there exists g ∈ G such that αg = β.
A group which does not act transitively is called intransitive. - 18Orbit-Stabiliser TheoremLet G be a group acting on a
set Ω. Then, for all α ∈ Ω,
|Gα||αG| = |G| - 19RegularA transitive action of G on Ω is called regular if Gα = 1 for all α ∈ Ω. Equivalently, g ∈ G fixes no point in Ω.
- 20Isomorphic ActionsLet G be a group which has an action on Ω and Ω'. We say these actions are isomorphic if there exists a bijection
φ : Ω → Ω' s.t for all α ∈ Ω, g ∈ G,
φ(αg) = φ(α)g - 21ImprimitiveLet G be a group acting transitively on a set Ω. Let Ḇ be a set of subsets B that partition Ω. If B ∩ Bg = ∅ or B for all
g ∈ G, B ∈ Ḇ, then we say that Ḇ is a system of imprimitivity, or a system of blocks.
The subsets B are called blocks.
Note that, we may always take Ḇ to be {Ω}, or {{α} : α ∈ Ω}. These are known as trivial systems.
A group G is imprimitive if there exists a non-trivial system of imprimitivity; G is primitive if no such system exists. - 22G-CongruenceAn equivalence relation ∼ is called a G-congruence if it is preserved under the action of the group. That is,
α ∼ β ⇐⇒ αg ∼ βg
for all g ∈ G. - 23Semidirect ProductLet N and H be two groups. Suppose H acts on N in such a way that it respects the multiplicative structure of N. That is, h ∈ H acts on N as an automorphism of N. We write this action exponentially: n↦nᵸ is an automorphism of N.
We define a group G = {(n, h) : n ∈ N, h ∈ H} with multiplication
(n, h)(m, g) := (nm^(h−1), hg)
We call G the semidirect product of N by H.
Let G be a group, N◅G and H ≤ G. Then, G is a semidirect product of N by H if G=NH and N∩H = 1 - 24Wreath ProductLet N = A × · · · × A (m times) be the direct
product of m copies of the group A. Let H be a group which acts on N by
permuting the m copies of A. We define G to be the semidirect product of
N by H. This is called the wreath product of A by H, A is called the base
group and G is written G = A≀H, or G = AwrH. - 25MaximalLet G be a group and H ≤ G a subgroup. Then H is maximal in G if there does not exists a subgroup K of G such that s.t
H <≠ K <≠ G - 26k-transitiveLet G be a group acting on a set Ω. Suppose α₁, . . . , α(k) are
k distinct points of Ω and β₁, . . . , β(k) are a second set of k distinct points of
Ω. Then, G is k-transitive if there exists g∈G s.t. α₁g = β₁, . . . , α(k)g = β(k)
If a group is k-transitive, then it is also (k − 1)-transitive. - 27SharplyA group G is sharply k-transitive if it is
k-transitive and the pointwise stabiliser of any k distinct points is trivial. - 28BaseA subset Σ ⊆ Ω is a base for G if G((Σ)) = 1. That is, if the only element which fixes Σ pointwise is the identity.
- 29SupportDefine supp(g) to be the set of points of Ω which are moved by g.
- 30Proper Primitive GroupA proper primitive group is a faithful primitive group of degree n which is not A(n), or S(n).
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- Fichas sobre Permutation Groups, creado por artibv el 02/05/2015.
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