- Understand the definition of a group.
- Learn the properties and axioms of groups.
- Explore examples of groups.
Definition of a Group
A group is a set \(G\) equipped with a binary operation \(*\) that combines any two elements \(a, b \in G\) to form another element \(a * b \in G\), and satisfies the following axioms:
- Closure: For all \(a, b \in G\), \(a * b \in G\).
- Associativity: For all \(a, b, c \in G\), \((a * b) * c = a * (b * c)\).
- Identity Element: There exists an element \(e \in G\) such that for all \(a \in G\), \(a * e = e * a = a\).
- Inverse Element: For each \(a \in G\), there exists an element \(b \in G\) such that \(a * b = b * a = e\), where \(e\) is the identity element.
Group Properties and Axioms
- Closure: The binary operation \(*\) must be closed in the group \(G\).
- Associativity: The group operation must be associative.
- Identity Element: There must be an identity element \(e\) in the group such that \(a * e = e * a = a\) for all \(a \in G\).
- Inverse Element: Every element \(a\) in the group must have an inverse \(b\) such that \(a * b = b * a = e\).
Examples of Groups
- Integers under Addition: The set of integers \(\mathbb{Z}\) with the operation of addition is a group. The identity element is \(0\), and the inverse of \(a\) is \(-a\).
- Non-zero Real Numbers under Multiplication: The set of non-zero real numbers \(\mathbb{R}^*\) with the operation of multiplication is a group. The identity element is \(1\), and the inverse of \(a\) is \(\frac{1}{a}\).
- Symmetric Group: The set of all permutations of a finite set \(S\) forms a group under the operation of composition. This group is called the symmetric group and is denoted by \(S_n\) for a set of \(n\) elements.
Exercises
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Verify Group Properties:
- Show that the set of integers \(\mathbb{Z}\) with addition is a group by verifying the closure, associativity, identity, and inverse properties.
- Show that the set of non-zero real numbers \(\mathbb{R}^*\) with multiplication is a group by verifying the closure, associativity, identity, and inverse properties.
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Examples of Groups:
- Find the identity element and inverse for the set of integers modulo \(n\), denoted by \(\mathbb{Z}_n\), with addition modulo \(n\).
- Show that the set of \(2 \times 2\) invertible matrices with real entries forms a group under matrix multiplication.
Summary
In this lesson, we introduced the concept of a group, defined its properties and axioms, and explored several examples of groups. Understanding the definition and properties of groups is fundamental to studying more advanced topics in abstract algebra.