Groups and Group Actions

Question 1

Define a binary operation

Answer 1

A binary operation on a set S is a function from SxS to S.

Question 2

Define associativity

Answer 2

If * is associative, a(bc) = (ab)c for all a, b and c in the set.

Question 3

Define an identity

Answer 3

e is an identity if ea = a = ae for all a in the set.

Question 4

Define an inverse

Answer 4

If b is in S, then b is a’s inverse if ab = ba = e.

Question 5

Define commutativity

Answer 5

If * is commutative, ab = ba for all a and b in the set

Question 6

Define a group

Answer 6

(G,*) is a group if:
1. * is associative
2. * has an identity e
3. Every a ∈ G has an inverse

Question 7

Define an Abelian group

Answer 7

A group that has a commutative binary operation.

Question 8

Define a cyclic group

Answer 8

A group G is cyclic if there is an element x ∈ G such that G = {x^n: n ∈ Z}.

Question 9

Define the dihedral group

Answer 9

D_2n is the set of all isometries of the regular n-gon in R^2, with the group operation being composition.

Question 10

Define a generator.

Answer 10

If there exists an element x ∈ G such that G = {x^n: n ∈ Z}, x is said to be a generator for G.

Question 11

Define a direct product

Answer 11

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.

Question 12

Define a subgroup

Answer 12

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.

Question 13

Define the order of a group

Answer 13

The cardinality of the underlying set.

Question 14

Gefine the order of an element

Answer 14

The order of g is the least positive integer r such that g^r = e.

Question 15

Define an isomorphism

Answer 15

If we can define a bijection that maps from one set to another, this is called an isomorphism.

Question 16

Define a permutation

Answer 16

A permutation of S is a bijection S -> S.

Question 17

Define a cycle

Answer 17

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}.

Question 18

Define disjoint cycles

Answer 18

Two cycles are disjoint if none of the elements in each cycle are shared between the two cycles.

Question 19

Define a cycle decomposition type

Answer 19

A list of the lengths of cycles in a permutation.

Question 20

Define conjugate permutations

Answer 20

Two permutations σ, τ ∈ S_n are conjugate if there exists ρ ∈ S_n such that (ρ^-1)τ ρ = σ.

Question 21

Define an odd permutation

Answer 21

A permutation is odd if it can be written as an odd number of transpositions.

Question 21

Define a transposition

Answer 21

A 2-cycle

Question 22

Define an even permutation

Answer 22

A permutation is even if it can be written as an even number of transpositions.

Question 23

Define the subgroup generated by S.

Answer 23

The smallest subgroup which contains S.

Question 24

Define the highest common factor

Answer 24

The highest common factor of m and n is h such that =

Question 25

Define the lowest common multiple

Answer 25

The lowest common multiple of m and n is l such that = ∩

Question 26

Define a binary relation

Answer 26

A binary relation on a set S is a subset of SxS

Question 26

Define a reflexive relation

Answer 26

~ is reflexive if a~a for all a in S

Question 27

Define a symmetric relation

Answer 27

~ is symmetric if a~b implies b~a for all a, b in S

Question 28

Define a transitive relation

Answer 28

~ is transitive if a~b and b~c implies that a~c for all a, b, c in S

Question 29

Define an Equivalence relation

Answer 29

~ is an equivalence relation if it is reflexive, symmetric and transitive

Question 30

Define conjugate elements

Answer 30

For a group G, g, h ∈ G are conjugate if there exists k∈G such that g = k^-1hk.

Question 31

Define an equivalence class

Answer 31

Given an equivalence relation ~, the equivalence class of a is the set {x∈S:x~a}.

Question 32

Define a partition

Answer 32

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 = ∅.

Question 33

Define a left coset

Answer 33

If H is a subgroup of G, then the left cosets of H are the sets gH = {gh:h∈H}.

Question 34

Define a right coset

Answer 34

If H is a subgroup of G, then the right cosets of H are the sets Hg = {hg:h∈H}.

Question 35

Define a homomorphism

Answer 35

φ: G -> H is a homomorphism if φ(g1*g2) = φ(g1)*φ(g2).

Question 36

Define an automorphism

Answer 36

An isomorphism from G to G.

Question 37

Define an endomorphism

Answer 37

A homomorphism from G to G.

Question 38

Define the kernel of a group

Answer 38

The kernel of φ is ker(φ) = {g∈G such that φ(g) = e}.

Question 39

Define the image of a group

Answer 39

The image of φ is Im(φ) = {φ(g) such that g∈G}.

Question 40

Define a normal subgroup

Answer 40

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.

Question 41

Define the centre of a group

Answer 41

Z(G) = {g∈G such that hg = gh for all h∈G}.

Question 42

Define a quotient group

Answer 42

If H is a normal subgroup of G, then (G/H,*) is the quotient group.

Question 43

Define a left action

Answer 43

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.

Question 44

Define an orbit

Answer 44

If a group G acts on a set S and s∈S, then Orb(s) := {g.s such that g∈G}.

Question 45

Define a stabilizer

Answer 45

If a group G acts on a set S and s∈S, then Stab(s) := {g∈G such that g.s = s}.