Subgroup Cosets (Left Coset, Right Coset) Definition

Introduction

In group theory, the concept of cosets arises naturally when considering the structure of a group and its subgroups. A coset is a set of elements formed by multiplying every element of a subgroup by a fixed element from the group. This notion is fundamental in understanding the relationship between subgroups and the overall structure of a group. Specifically, the study of left cosets and right cosets provides insight into the quotient groups and the index of a subgroup within a larger group.

A subgroup of a group $ G $ is a non-empty subset $ H \subseteq G $ that is closed under the group operations (multiplication and inversion) and satisfies the group axioms. The left coset of a subgroup $ H $ with respect to an element $ g \in G $ is defined as $ gH = { g h \mid h \in H } $. Similarly, the right coset is $ Hg = { h g \mid h \in H } $. These cosets partition the group $ G $, and their study reveals critical properties of group structures.

Core Content

Left Cosets

A left coset is a set of elements generated by multiplying each element of the subgroup $ H $ with a fixed element $ g \in G $. Formally, for any $ g \in G $, the left coset $ gH $ consists of all products $ g h $ where $ h \in H $. The set $ gH $ is a subset of $ G $, and the collection of all left cosets of $ H $ forms a partition of $ G $, provided that $ H $ is a subgroup of $ G $.

The size of each left coset is equal to the order of the subgroup $ H $, i.e., $ |gH| = |H| $. This property is crucial in understanding the index of a subgroup, denoted $ [G : H] $, which is the number of distinct left cosets. The index is defined as $ [G : H] = \frac{|G|}{|H|} $, and it must be an integer for $ H $ to be a subgroup of $ G $.

Right Cosets

The concept of right cosets mirrors that of left cosets. For any $ g \in G $, the right coset $ Hg $ is the set of all products $ h g $ where $ h \in H $. The right cosets also partition $ G $, and their number is the same as the number of left cosets. This symmetry between left and right cosets is particularly evident when $ H $ is a normal subgroup of $ G $, in which case the left and right cosets coincide.

Properties of Cosets

1. Partitioning Property: The left and right cosets of a subgroup $ H $ partition the group $ G $. That is, $ G = \bigcup_{g \in G} gH $ and $ G = \bigcup_{g \in G} Hg $.
2. Equivalence Classes: Each left coset $ gH $ is equivalent to a right coset $ Hg $ under the relation $ gH = Hg $, which holds if and only if $ H $ is a normal subgroup.