A **subgroup** is a subset \( H \) of a group \( G \) that itself forms a group under the same operation as \( G \). For \( H \) to be a subgroup of \( G \), it must satisfy the following three conditions:

1. **Closure**: For every \( a, b \in H \), the product \( a \cdot b \in H \).
2. **Identity**: The identity element of \( G \) must be in \( H \).
3. **Inverses**: For every \( a \in H \), its inverse \( a^{-1} \in H \).

### Examples of Subgroups:
1. **Trivial Subgroup**: Every group \( G \) has the trivial subgroup \( \{e\} \), where \( e \) is the identity element of \( G \).
   
   Example: In \( G = \mathbb{Z} \) (the group of integers under addition), the trivial subgroup is \( \{0\} \).
   
2. **Whole Group**: The group itself \( G \) is always a subgroup of \( G \).
   
   Example: \( \mathbb{Z} \), the group of integers under addition, is a subgroup of itself.

3. **Subgroup of Integers**: \( n\mathbb{Z} \), the set of multiples of an integer \( n \), forms a subgroup of \( \mathbb{Z} \).
   
   Example: \( 2\mathbb{Z} = \{ \dots, -4, -2, 0, 2, 4, \dots \} \) is a subgroup of \( \mathbb{Z} \).

4. **Subgroup of Symmetric Group**: The alternating group \( A_n \) (the group of even permutations) is a subgroup of the symmetric group \( S_n \) (the group of all permutations) on \( n \) elements.
   
   Example: \( A_3 = \{e, (123), (132)\} \) is a subgroup of \( S_3 \), the symmetric group on 3 elements.

5. **Subgroup of Real Numbers**: The set of positive real numbers \( \mathbb{R}^+ \) under multiplication is a subgroup of \( \mathbb{R}^\times \) (the set of all non-zero real numbers under multiplication).

   Example: \( \mathbb{R}^+ = \{ x \in \mathbb{R} \mid x > 0 \} \) is a subgroup of \( \mathbb{R}^\times \).

Would you like more examples or details on specific properties of subgroups?

### Related Questions:
1. What is a normal subgroup, and how does it differ from a general subgroup?
2. How can we prove whether a given set is a subgroup?
3. What is Lagrange's theorem in the context of subgroups?
4. Can you explain cyclic subgroups with examples?
5. What are the intersections and unions of subgroups?

**Tip**: Every group has at least two subgroups: the trivial subgroup and the group itself!

Explore the definition of subgroups in group theory with examples such as the trivial subgroup, whole group, and subgroups of integers. This concept involves key theorems like Lagrange's theorem and the subgroup criterion, ideal for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation