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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.

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