1 Operations

Here we review, with references but without great detail, the basic definitions and results that facilitate our study of representation theory.

Definition 1.1 A binary operation on a set \(X\) is a map \(\ast: X \times X \to X\). We say that \(\ast\) is:

unital if there exists an element \(e \in X\) such that \(e \ast x = x = x \ast e\) for all \(x \in X\).
associative if \((x \ast y) \ast z = x \ast (y \ast z)\) for every \(x, y, z \in X\).
commutative if \(x \ast y = y \ast x\) for every \(x, y \in X\).

Example 1.1 (Arithmetic operations) We are accustomed to operations on \(\mathbb{Z}, \mathbb{Q}, \mathbb{R},\) and \(\mathbb{C}\):

Addition (\(+\)) and multiplication (\(\cdot\)) are unital, commutative, and associative binary operations on these sets.
Subtraction is a (non-unital, non-associative, non-commutative) binary operation on these sets.
Division is a (non-unital, non-associative, non-commutative) binary operation on the sets \(\mathbb{Q}-\{0\}, \mathbb{R}-\{0\}, \mathbb{C}-\{0\}\), but not on \(\mathbb{Z}-\{0\}\).

Example 1.2 (Matrix operations) If we write \(\operatorname{Mat}_n(\mathbb{R})\) for the set of \(n \times n\) matrices with real entries, or more generally \(\operatorname{Mat}_n(F)\) with respect to some arbitrary field \(F\), then:

Matrix addition is a unital, commutative, and associative binary operation.
Matrix multiplication is a unital and associative, but non-commutative, binary operation.

Example 1.3 (Function composition) If \(X\) is a set, we write \(\operatorname{Func}(X,X)\) to denote the set of functions \(X \to X\). The operation of composition, i.e., assigning \(f, g \in \operatorname{Func}(X,X)\) to the map \(f \circ g: X \to X\) which sends \(x \mapsto f(g(x))\), is associative and unital but generally non-commutative.

Example 1.4 (Cross product) The standard cross-product \(\times\) on \(\mathbb{R}^3\) is a non-unital, non-commutative, non-associative binary operation.

Example 1.5 (Rock-paper-scissors) The following operation on the set \(\{r,p,s\}\) is commutative, but non-associative and non-unital:

\(∗\)	\(r\)	\(p\)	\(s\)
\(r\)	\(r\)	\(p\)	\(r\)
\(p\)	\(p\)	\(p\)	\(s\)
\(s\)	\(r\)	\(s\)	\(s\)

In particular, \(s = p \ast s = (r \ast p) \ast s \not = r \ast (p \ast s) = r \ast s = r\).

Proposition 1.1 (Uniqueness of identity) If \(\ast\) is a unital binary operation on a set \(X\) with identity \(e\), and there is an element \(e' \in X\) satisfying \(e' \ast x = x\) for all \(x \in X\), then \(e' = e\).

Definition 1.2 Let \(\ast\) be a unital binary operation on a set \(X\) and fix \(x \in X\). If there is an element \(y \in X\) such that \(x \ast y = e = y \ast x\), then we say that \(x\) is invertible with inverse \(y\).

Proposition 1.2 (Uniqueness of inverses) If \(\ast\) is an associative unital binary operation on a set \(X\) and \(x,y,z \in X\) satisfy \(x \ast y = e = x \ast z\), then \(y=z\).

Example 1.6 (Negatives) Every element in \(\mathbb{Z}, \mathbb{Q}, \mathbb{R},\) and \(\mathbb{C}\) is invertible with respect to \(+\).

Example 1.7 (Fractions) The element \(2 \coloneqq 1+1\) is invertible with respect to \(\cdot\) in \(\mathbb{Q}\) but not \(\mathbb{Z}\). Indeed, every non-zero element of \(\mathbb{Q}, \mathbb{R},\) and \(\mathbb{C}\) is invertible with respect to \(\cdot\).

Example 1.8 (Invertible functions) A function \(f \in \operatorname{Func}(X,X)\) is called a permutation of \(X\) if it is a bijection; we write \(\operatorname{Perm}(X) \subseteq \operatorname{Func}(X,X)\) for the set of all permutations of \(X\). These are exactly the invertible elements (with respect to composition) in Example 1.3.

Definition 1.3 (Groups) (Dummit and Foote 2003, 16) A group is a set \(G\) together with a unital associative binary operation \(\ast\), such that every \(g \in G\) is invertible. If \(\ast\) is commutative, then \(G\) is called an abelian group.¹

Definition 1.4 (Rings) (Dummit and Foote 2003, 223) A set \(R\) equipped with a pair of binary operations, usually named addition (\(+\)) and multiplication (\(\cdot\)), is called a ring if:

\(R\) is an abelian group with respect to addition.
multiplication is unital² and associative.
distributive laws hold: \[ (x+y) \cdot z = x \cdot z + y \cdot z \quad \text{ and } \quad x \cdot (y+z) = x \cdot y + x \cdot z, \] for any \(x,y,z \in R\).

The identity of \(+\) is called \(0_R\) while the identity of \(\cdot\) is written \(1_R\), or simply as \(0\) and \(1\) if context is clear. If multiplication is commutative, we say that \(R\) is a commutative ring.

Definition 1.5 (Fields) If \(F\) is a commutative ring in which every non-zero element is invertible with respect to multiplication, then we say that \(F\) is a field. Equivalently, \(F\) is a field if the set \(F-\{0\}\) is a group with respect to multiplication.

This word is in reference to the early 19th century mathematician, Niels Abel, who proved that polynomials whose Galois groups are commutative can always be solved in some precise sense.↩︎
In some books—e.g., Dummit and Foote (2003)—rings are not required to contain a multiplicative identity. This is a matter of taste.↩︎