Skip to main content

Subgroups and Cyclic Groups

  • Understand the definition and properties of subgroups.
  • Learn about cyclic groups and their characteristics.
  • Explore examples of subgroups and cyclic groups.

Definition of Subgroups

A subgroup \( H \) of a group \( G \) is a subset of \( G \) that itself forms a group under the same operation as \( G \).

Formal Definition: A subset \( H \subseteq G \) is a subgroup of \( G \) if:

  • Closure: For all \( a, b \in H \), \( a * b \in H \).
  • Identity: The identity element of \( G \), \( e \), is in \( H \).
  • Inverses: For every \( a \in H \), the inverse \( a^{-1} \) is also in \( H \).

Notation: If \( H \) is a subgroup of \( G \), it is denoted as \( H \leq G \).

Properties of Subgroups

  • Closure: The operation within the subgroup remains closed.
  • Identity: The subgroup contains the identity element of the group.
  • Inverse: Every element in the subgroup has an inverse that is also in the subgroup.
  • Associativity: Inherited from the parent group \( G \).

Definition of Cyclic Groups

A cyclic group is a group that can be generated by a single element. This means every element of the group can be written as powers (or multiples) of this generator.

Formal Definition: A group \( G \) is cyclic if there exists an element \( g \in G \) such that every element in \( G \) can be expressed as \( g^n \) for some integer \( n \).

Notation: If \( G \) is a cyclic group generated by \( g \), it is denoted as \( \langle g \rangle \).

Properties of Cyclic Groups

  • Generator: A single element can generate the entire group.
  • Finite Cyclic Groups: If \( G \) is finite, the order of \( G \) is the smallest positive integer \( n \) such that \( g^n = e \).
  • Infinite Cyclic Groups: If \( G \) is infinite, \( G \) is isomorphic to \( \mathbb{Z} \) (the group of integers under addition).

Examples of Subgroups and Cyclic Groups

  • Subgroup Example: Consider the group \( (\mathbb{Z}, +) \) (integers under addition). The set of even integers \( 2\mathbb{Z} \) is a subgroup of \( \mathbb{Z} \).
  • Cyclic Group Example: The group of integers modulo \( n \), \( \mathbb{Z}_n \), is a cyclic group. For example, \( \mathbb{Z}_6 \) (integers modulo 6) is cyclic and can be generated by 1 or 5.
  • Symmetric Group: The symmetric group \( S_3 \) has subgroups that are also cyclic, such as the subgroup generated by a single transposition.

Exercises

  • Identify Subgroups: Given the group \( (\mathbb{Z}, +) \), identify at least three different subgroups.
  • Generate Cyclic Groups: Show that \( \mathbb{Z}_4 \) is a cyclic group and identify all possible generators.
  • Subgroup Verification: Verify that the set of all \( 2 \times 2 \) invertible matrices with real entries forms a subgroup under matrix multiplication.

Summary

In this lesson, we explored the concepts of subgroups and cyclic groups, including their definitions, properties, and examples. Understanding these fundamental ideas is crucial as they form the building blocks for more advanced topics in group theory and abstract algebra.

This Week's Best Picks from Amazon

Please see more curated items that we picked from Amazon here .

Popular posts from this blog

LU Decomposition

LU Decomposition: A Step-by-Step Guide LU Decomposition, also known as LU Factorization, is a method of decomposing a square matrix into two triangular matrices: a lower triangular matrix L and an upper triangular matrix U . This is useful for solving linear equations, computing determinants, and inverting matrices efficiently. What is LU Decomposition? LU Decomposition expresses a matrix A as: \[ A = LU \] where: L is a lower triangular matrix with ones on the diagonal. U is an upper triangular matrix. Step-by-Step Process Consider the matrix: \[ A = \begin{bmatrix} 2 & 3 & 1 \\ 4 & 7 & 3 \\ 6 & 18 & 5 \end{bmatrix} \] Step 1: Initialize L as an Identity Matrix Start with an identity matrix for \( L \): \[ L = \begin{bmatrix} 1 & 0 & 0 \\ 0 ...
Read more

Gaussian Elimination: A Step-by-Step Guide

Gaussian Elimination: A Step-by-Step Guide Gaussian Elimination is a systematic method for solving systems of linear equations. It works by transforming a given system into an equivalent one in row echelon form using a sequence of row operations. Once in this form, the system can be solved efficiently using back-substitution . What is Gaussian Elimination? Gaussian elimination consists of two main stages: Forward Elimination: Convert the system into an upper triangular form. Back-Substitution: Solve for unknowns starting from the last equation. Definition of a Pivot A pivot is the first nonzero entry in a row when moving from left to right. Pivots are used to eliminate the elements below them, transforming the system into an upper triangular form. Step-by-Step Example Consider the system of equations: \[ \begin{aligned} 2x + 3y - z &= 5 \\ 4x + y...
Read more

Vector Spaces and Linear Transformation

Vector Spaces and Linear Transformations A vector space is a set of vectors that satisfies specific properties under vector addition and scalar multiplication. Definition of a Vector Space A set \( V \) is called a vector space over a field \( \mathbb{R} \) (real numbers) if it satisfies the following properties: Closure under addition: If \( \mathbf{u}, \mathbf{v} \in V \), then \( \mathbf{u} + \mathbf{v} \in V \). Closure under scalar multiplication: If \( \mathbf{v} \in V \) and \( c \in \mathbb{R} \), then \( c\mathbf{v} \in V \). Associativity: \( (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) \). Commutativity: \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \). Existence of a zero vector: There exists a vector \( \mathbf{0} \) such that \( \mathbf{v} + \mathbf{0} = \mathbf{v} \). Existence of additive inverses: For eac...
Read more