1.1  Axioms and Examples

---

Definition 1.1.1.

1. A binary operation \(\ast\) on a set \(G\) is a function \(\ast:G\times G\to G\). We shall write \(a\ast b\) for \(\ast(a,b)\).

2. A binary operation \(\ast\) on \(G\) is said to be associative if

   \[a\ast (b\ast c) = (a\ast b)\ast c\]

   for all \(a,b,c,\in G\).

3. Let \(\ast\) be a binary operation on \(G\). The elements \(a\) and \(b\) of \(G\) commute if

   \[a\ast b = b\ast a.\]

   The binary operation \(\ast\) is said to be commutative if \(a\) and \(b\) commute for all \(a,b\in G\).

Definition 1.1.2. Suppose that \(\ast\) is a binary operation on a set \(G\), and \(H\) is a subset of \(G\). If the restriction of \(\ast\) to \(H\) is a binary operation on \(H\), that is, \(a\ast b\in H\) for all \(a,b\in H\), then \(H\) is said to be closed under \(\ast\).

Note that if \(\ast\) is associative (respectively, commutative) on \(G\), and if \(H\) is closed under \(\ast\), then \(\ast\) is also associative (respectively, commutative) on \(H\).

Definition 1.1.3. A group is an ordered pair \((G,\ast)\) where \(G\) is a set and \(\ast\) is a binary operation on \(G\) satisfying the following axioms:

1. \(\ast\) is associative;

2. there is \(e\in G\), called an identity of \(G\), such that

   \[a\ast e = e \ast a = a\]

   for all \(a\in G\);

3. for each \(a\in G\) there is \(a^{-1}\in G\), called the an inverse of \(a\), such that

   \[a\ast a^{-1} = a^{-1} \ast a = e.\]

The group \((G,\ast)\) is said to be abelian if \(\ast\) is commutative. Less formally, we shall often say that \(G\) is a group under \(\ast\). Also, we say \(G\) is a finite group if in addition \(G\) is finite.

References

1. Dummit, D. S., & Foote, R. M. (2004). Abstract algebra (3rd ed.). Wiley.

---