Definitions

Introduction to Group Theory Series

Part I

1. A binary operation • on a set S is a function that sends pairs of elements of S to single elements of S. We write •(a, b) = c as a • b = c.

2. A binary operation • on a set S is closed if, for all a, b in S, a • b is also in S.

3. Let (S, •) be a set equipped with a binary operation. We say that e ∈ S is an identity element if, for all s ∈ S,

   \(e \cdot s = s \cdot e = s.\)

4. Let (S, •) be a set equipped with a binary operation. We say that an element b ∈ S is the inverse of an element a ∈ S if

   \(a \cdot b = b \cdot a = e.\)

5. A binary operation • on a set S is associative if, for all a, b, c ∈ S,

   \((a \cdot b) \cdot c = a \cdot (b \cdot c).\)

6. A binary operation • on a set S is commutative if, for all a, b ∈ S,

   \(a \cdot b = b \cdot a.\)

7. A group is a pair (G, •) such that G is a set, • is a binary operation that is closed on G, and the following three properties hold:

   1. The binary operation • is associative on G;

   2. There exists a unique identity element e ∈ G;

   3. For every g ∈ G, there exists a unique inverse g⁻¹ ∈ G of g.

8. A group G is called Abelian if the binary operation • is commutative on G.

Part II

1. A function f : A → B is, informally, a way of assigning elements of A to elements of B. (Formally, f is a set of pairs (a, b), where a ∈ A and b ∈ B, such that every element of A is in exactly one pair.) We write f(a) = b if the pair (a, b) is in our set. We call the set A the domain of the function, and B the codomain of the function.

2. A function f : A → B is injective/is an injection if every element of the domain gets mapped to a different element of the codomain. One way of saying this is that

   \(f(a) = f(b) \quad \text{implies} \quad a = b.\)

3. A function f : A → B is surjective/is a surjection if every element of the codomain is an output of f. That is, for any b ∈ B, there exists an a ∈ A such that f(a) = b.

4. A function f : A → B is bijective/is a bijection if it is both an injection and a surjection. This implies that f has an inverse function, which we denote f⁻¹. The inverse function is defined so that, if

   \(f(a) = b, \quad \text{then} \quad f^{-1}(b) = a.\)

   Note that f⁻¹ : B → A, not f⁻¹ : A → B.

Part III

1. A subgroup of a group G is a set H ⊆ G (this ⊆ symbol just means H is contained in G - think of it like ≤ but for sets), that is closed under • and is itself a group. We call G an improper subgroup of itself - all other subgroups are proper subgroups. Likewise, the group {e} that contains only the identity element is called the trivial group, and is considered to be a trivial subgroup of G. All other subgroups are called non-trivial.

Part IV

1. Let G and H be groups. A function φ : G → H is called a homomorphism if, for all a and b in G, we have

   \(\phi(ab) = \phi(a)\phi(b).\)

   If φ is also a bijection, we say that φ is an isomorphism. If there is an isomorphism between two groups G and H, we say they are isomorphic and write

   \(G \cong H.\)

Part V

1. Let ϕ : X → Y be a function, and suppose A ⊆ X and B ⊆ Y (that is, A is a subset of X, and B is a subset of Y). We define

   \(\phi[A] = \{\phi(a) \: | \: a \in A\}\)

   and call it the image of A under ϕ. Likewise, we define

   \(\phi^{-1}[B] = \{a \in A \: | \: \phi(a) \in B\}\)

   and call it the preimage or inverse image of B under ϕ.

2. Let ϕ : G → G' be a group homomorphism. The subgroup

   \(\phi^{-1}[\{e'\}] = \{x \in G \: | \: \phi(x) = e'\}\)

   is called the kernel of ϕ, which we denote ker(ϕ).

Part VI

1. Let G be a group and H a subgroup of G. The subset

   \(aH = \{ah \: | \: h \in H\}\)

   is called the left coset of H containing a. Similarly, the set

   \(Ha = \{ha \: | \: h \in H\}\)

   is called the right coset of H containing a.

2. Let H be a subgroup of G. The number of cosets of H in G is called the index of H in G, which we denote (G : H).

3. Let H be a subgroup of G. We say that H is a normal subgroup of G if for all g ∈ G, gH = Hg. We denote this by H ⊴ G.