Group Theory Definitions

Binary operation.

A binary operation on a set S is a function  δ: S X S →S. Notation: We often write s ⊕ t, s ⊗ t, or st in place of δ(s, t).

Cyclic subgroup.

A cyclic subgroup is a subgroup that, considered as a group, is cyclic. That is, H =< g > for some g ∈ H, and we say that H is the subgroup generated by g.

One-to-one.

A function f : A → R is one-to-one if f(a) = f(b) implies that a = b.

Onto.

A function f : A → R is onto if for every y ∈ R, there exists a ∈ A such that f(a) = y.

Abelian.

A group G is abelian if if gh = hg for all g;,h ∈ G. (The group operation is commutative.)

Cyclic.

A group G is cyclic if there is an element g ∈ G such that < g >= G. In this case, g is called a generator for G.

Partition.

A partition of a set X is a collection {X₀} of subsets of X such that (1) every element of X is in some X₀; (2) if X₀ ≠ X₁, then X₀ ∩ X₁= ∅;. (That is, any two subsets are either equal or disjoint.)

Permutation.

A permutation of set X is a map α : X → X such that α is one-to-one and onto.

Equivalence Relation.

A relation on a set S is a set R of pairs of elements of S; that is, R ⊆ S X S. We often use the symbol ∼ for the relation, and write s ∼ t to mean (s, t) ∈ R. A relation  on S is an equivalence relation if it has the following properties. (1) For all s ∈ S, s ∼ s. (The relation is reflexive.) (2) If s ∼ t, then t ∼ s. (The relation is symmetric.) (3) If s ∼ t and t ∼ r then s ∼ r. (The relation is transitive.)

Symmetry.

A symmetry of a geometric object S is a rigid motion in 3-space that takes S to itself.

Euler's phi function.

Euler's phi function δ : Z+ → n is defined by δ(n) equals the number of positive integers less than n that are relatively prime to n.

Equivalence class.

Let ∼ be an equivalence relation on a set S. For a ∈ S, the equivalence class for ∼ containing a is a(with a bar over it, abar) = {s ∈ S : a ∼ s}.

Kernel.

Let δ : G → H be a homomorphism of groups. The kernel of δ is {g ∈ G : δ(g) = 1}; this is denoted ker(δ).

Direct product of groups.

Let G and H be groups, and use 1G for the identity of G and 1H for the identity of H. The direct product of G and H is the group with set G X H = {(g, h) : g ∈ G, h ∈ H} and operation defined by (g, h)(a, b) = (ga, hb). (Note: Problem 29 shows that this is a group.)

Isomorphism.

Let G and H be groups. A function f : G → H is an isomorphism if  is one-to-one, onto, and for all a, b ∈ G, f(ab) = f(a)f(b).

Homomorphism.

Let G and H be groups. A function δ : G → H is an homomor- phism if for all a, b ∈ G, δ(ab) = δ(a)δ(b).

Isomorphic.

Let G and H be groups. G is isomorphic to H if there exists an isomorphism from G to H. Notation: G ≅ H means G is isomorphic to H.

Coset.

Let G be a group and H a subgroup of G. For a ∈ G, define aH = {ah : h ∈ H], and Ha = {ha : h ∈ H]. The left cosets of H in G are the sets aH with a ∈ G. The right cosets of H in G are the sets Ha with a ∈ G.

Normal subgroup.

Let G be a group and H a subgroup of G. H is a normal sub- group of G if gH = Hg for all g ∈ G.

Quotient group.

Let G be a group, and H a normal subgroup of G. The quotient group of G modulo H is the group whose set of elements is the set of left cosets of H, with operation given by aHbH = abH. Notation: G / H denotes the quotient group G mod H.

Index of a subgroup.

Let G be a group, and H a subgroup of G. The index of H in G is the number of left cosets of H in G. Note: this is equal to the number of right cosets of H in G. Notation: |G : H| is the index of H in G.

Subgroup.

Let G be a group. A subset H is a subgroup of G if H, together with the binary operation inherited from G, is itself a group. Notation: H ≤ G means that H is a subgroup of G. H < G means that H is a subgroup of G but H ≠ G. These are also sometimes written as G ≥ H and G > H.

Order.

Let G be a group. The order of G is the number of elements in G. Note that this can be infinite (∞). Notation: |G| is the order of G.

Group.

Let G be a set and  a binary operation on G. We say that (G,⊕ ) is a group, if the following axioms hold. 1. For all g, h, k ∈ G, (g ⊕ h) ⊕ k = g ⊕ (h ⊕ k). (The group operation is associative.) 2. There exists an element e ∈ G such that for all g ∈ G, e ⊕ g = g ⊕ e = g. e is called the identity of the group. 3. For every g ∈ G, there exists an element x ∈ G such that g ⊕ x = x ⊕ g = e. The element x is called an inverse of g.

Center.

The center of a group G is the set Z(G) = {a ∈ G : ag = ga for all g ∈ G}.

Order of an element.

The order of an element g in G is the smallest positive integer n such that gⁿ = 1. If there is no such n, then the order of g is ∞. Notation: o(g) denotes the order of g.