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Cosets and Lagrange's Theorem

  • Understand the concept of cosets.
  • Learn about left and right cosets.
  • Explore Lagrange's Theorem and its implications.

Definition of Cosets

Given a group \( G \) and a subgroup \( H \) of \( G \), a coset is a form of subset that can be defined as follows:

Left Coset: For any element \( g \in G \), the left coset of \( H \) with respect to \( g \) is the set \( gH = \{ gh \mid h \in H \} \).

Right Coset: For any element \( g \in G \), the right coset of \( H \) with respect to \( g \) is the set \( Hg = \{ hg \mid h \in H \} \).

Left and Right Cosets

Cosets can be classified into two types:

  • Left Cosets: A left coset of \( H \) in \( G \) is a set formed by multiplying \( H \) with a fixed element from the left.
  • Right Cosets: A right coset of \( H \) in \( G \) is a set formed by multiplying \( H \) with a fixed element from the right.

Note that while left cosets and right cosets may differ, they both partition the group \( G \) into disjoint subsets of equal size.

Lagrange's Theorem

Lagrange's Theorem is a fundamental result in group theory that relates the order of a subgroup to the order of the entire group.

Theorem: If \( G \) is a finite group and \( H \) is a subgroup of \( G \), then the order of \( H \) (the number of elements in \( H \)) divides the order of \( G \) (the number of elements in \( G \)).

Implication: The number of distinct left cosets (or right cosets) of \( H \) in \( G \) is equal to the index of \( H \) in \( G \), denoted as \( |G : H| = \frac{|G|}{|H|} \).

Proof of Lagrange's Theorem

Consider the group \( G \) and its subgroup \( H \). The left cosets of \( H \) in \( G \) are \( gH \) for \( g \in G \).

First, we show that the left cosets of \( H \) partition \( G \):

  • Every element \( g \in G \) belongs to some left coset \( gH \).
  • Two left cosets \( g_1H \) and \( g_2H \) are either identical or disjoint.

Next, we show that each left coset \( gH \) has the same number of elements as \( H \). This is because the function \( f_h: H \to gH \) defined by \( f_h(h) = gh \) is a bijection.

Since the left cosets partition \( G \) and each coset has the same number of elements as \( H \), the order of \( G \) is the product of the number of cosets and the order of \( H \):

\( |G| = |H| \times \text{number of distinct left cosets} \)

Thus, the order of \( H \) divides the order of \( G \), completing the proof.

Examples of Cosets and Lagrange's Theorem

  • Example 1: Consider the group \( \mathbb{Z}_6 \) (integers modulo 6) and its subgroup \( \{0, 3\} \). The left cosets of this subgroup in \( \mathbb{Z}_6 \) are \( \{0, 3\} \), \( \{1, 4\} \), and \( \{2, 5\} \).
  • Example 2: For the symmetric group \( S_3 \), consider the subgroup \( H \) generated by the identity permutation and a transposition. Lagrange's Theorem tells us that the order of \( H \) (which is 2) divides the order of \( S_3 \) (which is 6).

Exercises

  • Identify Cosets: Given the group \( \mathbb{Z}_{10} \) (integers modulo 10) and its subgroup \( \{0, 5\} \), identify all distinct left cosets.
  • Apply Lagrange's Theorem: For the group \( \mathbb{Z}_{12} \) and its subgroup \( \{0, 4, 8\} \), use Lagrange's Theorem to determine the number of distinct cosets.
  • Coset Verification: Verify that the left cosets and right cosets of the subgroup \( \{0, 2, 4\} \) in \( \mathbb{Z}_6 \) partition the group into disjoint subsets.

Summary

In this lesson, we explored the concepts of cosets and Lagrange's Theorem, including their definitions, properties, and examples. Understanding these fundamental ideas is crucial as they provide insights into the structure of groups and their subgroups.

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