Groups and Group Actions Flashcards
Define a binary operation
A binary operation on a set S is a function from SxS to S.
Define associativity
If * is associative, a(bc) = (ab)c for all a, b and c in the set.
Define an identity
e is an identity if ea = a = ae for all a in the set.
Define an inverse
If b is in S, then b is a’s inverse if ab = ba = e.
Define commutativity
If * is commutative, ab = ba for all a and b in the set
Define a group
(G,*) is a group if:
1. * is associative
2. * has an identity e
3. Every a ∈ G has an inverse
Define an Abelian group
A group that has a commutative binary operation.
Define a cyclic group
A group G is cyclic if there is an element x ∈ G such that G = {x^n: n ∈ Z}.
Define the dihedral group
D_2n is the set of all isometries of the regular n-gon in R^2, with the group operation being composition.
Define a generator.
If there exists an element x ∈ G such that G = {x^n: n ∈ Z}, x is said to be a generator for G.
Define a direct product
If we have two groups, (G,) and (G,o), then GxH (the direct product) is the set {(g,h): g ∈ G, h ∈ H} with group operation (g_1,h_1)”“(g_2,h_2) = (g_1*g_2,h_1oh_2) for all g_1, g_2 in G and h_1, h_2 in H.
Define a subgroup
H is a subgroup of G if H is a subset of G, e ∈ H, if x,y ∈ H then xy ∈ H and if x ∈ H, then x^-1 ∈ H.
Define the order of a group
The cardinality of the underlying set.
Gefine the order of an element
The order of g is the least positive integer r such that g^r = e.
Define an isomorphism
If we can define a bijection that maps from one set to another, this is called an isomorphism.
Define a permutation
A permutation of S is a bijection S -> S.
Define a cycle
A permutation σ ∈ S_n is a cycle if there are distinct elements a_1, … , a_k in {1, …, n} such that (a_i)σ = a_(i+1) for 1 ≤ i < k, (a_k)σ = a_1 and xσ = x for x ∉ {a_1, …, a_k}.
Define disjoint cycles
Two cycles are disjoint if none of the elements in each cycle are shared between the two cycles.
Define a cycle decomposition type
A list of the lengths of cycles in a permutation.
Define conjugate permutations
Two permutations σ, τ ∈ S_n are conjugate if there exists ρ ∈ S_n such that (ρ^-1)τ ρ = σ.
Define an odd permutation
A permutation is odd if it can be written as an odd number of transpositions.
Define a transposition
A 2-cycle
Define an even permutation
A permutation is even if it can be written as an even number of transpositions.
Define the subgroup generated by S.
The smallest subgroup which contains S.
Define the highest common factor
The highest common factor of m and n is h such that <h> = <m,n></h>
Define the lowest common multiple
The lowest common multiple of m and n is l such that <l> = <m>∩<n></n></m></l>
Define a binary relation
A binary relation on a set S is a subset of SxS
Define a reflexive relation
~ is reflexive if a~a for all a in S
Define a symmetric relation
~ is symmetric if a~b implies b~a for all a, b in S
Define a transitive relation
~ is transitive if a~b and b~c implies that a~c for all a, b, c in S
Define an Equivalence relation
~ is an equivalence relation if it is reflexive, symmetric and transitive
Define conjugate elements
For a group G, g, h ∈ G are conjugate if there exists k∈G such that g = k^-1hk.
Define an equivalence class
Given an equivalence relation ~, the equivalence class of a is the set {x∈S:x~a}.
Define a partition
Let S be a set and Λ be an indexing set. For λ∈Λ, A_λ partitions S if A_λ ≠ ∅ for all λ∈Λ, the union of A_λ = S and if λ≠µ, then A_λ∩A_µ = ∅.
Define a left coset
If H is a subgroup of G, then the left cosets of H are the sets gH = {gh:h∈H}.
Define a right coset
If H is a subgroup of G, then the right cosets of H are the sets Hg = {hg:h∈H}.
Define a homomorphism
φ: G -> H is a homomorphism if φ(g1g2) = φ(g1)φ(g2).
Define an automorphism
An isomorphism from G to G.
Define an endomorphism
A homomorphism from G to G.
Define the kernel of a group
The kernel of φ is ker(φ) = {g∈G such that φ(g) = e}.
Define the image of a group
The image of φ is Im(φ) = {φ(g) such that g∈G}.
Define a normal subgroup
H is a normal subgroup of G if H is a subgroup of G and gH = Hg for all g, or equivalently g^-1hg ∈ H for all g∈G and h∈H.
Define the centre of a group
Z(G) = {g∈G such that hg = gh for all h∈G}.
Define a quotient group
If H is a normal subgroup of G, then (G/H,*) is the quotient group.
Define a left action
ρ: GXS -> S is a left action if ρ(e,s) = s and ρ(g,ρ(h,s)) = ρ(gh,s) for all s∈S and g,h∈G.
Define an orbit
If a group G acts on a set S and s∈S, then Orb(s) := {g.s such that g∈G}.
Define a stabilizer
If a group G acts on a set S and s∈S, then Stab(s) := {g∈G such that g.s = s}.
