Group

Definition b. Let \(A_1, \ldots, A_n\) be sets. The Cartesian product \(A_1 \times \cdots \times A_n\) is the set of all ordered \(n\)-tuples \((a_1, \ldots, a_n)\) such that \(a_i \in A_i\) for \(1 \le i \le n\).

Definition n. Let \(A\) and \(B\) be sets. A map is a set \(\varphi \subseteq A \times B\) such that for each \(a \in A\) there is exactly one \(b \in B\) with \((a, b) \in \varphi\).

Definition d. Let \(A\) be a set. A binary operation is a map from \(A \times A\) to \(A\).

Definition r. A monoid is a set \(M\) with a binary operation \((a, b) \mapsto a \circ b\) that satisfies axioms 1. and 2. below. A monoid is a commutative monoid if axiom 3. also holds.

Associativity: \((a \circ b) \circ c = a \circ (b \circ c)\) for all \(a, b, c \in M\).
Identity: There is an identity element \(e \in M\) such that \(e \circ a = a \circ e = a\) for all \(a \in M\).
Commutativity: \(a \circ b = b \circ a\) for all \(a, b \in M\).

Remark k.

Since \(\circ\) is a binary operation, the resulting element, \(a \circ b\) is always in \(M\) for all \(a, b \in M\). We say that \(M\) is closed under \(\circ\) or that \(\circ\) is closed on \(M\).
For simplicity, we usually refer to the set \(M\) as the monoid with the associated operation being implicit. We use this convention for other algebraic structures as well, even when there are multiple operations associated with the structure.
We usually refer to the identity element as the identity.

Definition f. A group \(G\) is a monoid in which all \(a \in G\) have an inverse element \(b \in G\) with \(a \circ b = b \circ a = e\). A group \(G\) is an Abelian group if it is a commutative monoid.

Note that Abelian groups are commutative groups.