Left Coset

Definition

A left coset of a subgroup in a group is the set formed by multiplying each element of the subgroup by a fixed element from the group on the left. This concept is crucial in understanding how groups can be partitioned and helps illustrate the structure of groups and their subgroups, particularly when discussing normal subgroups and factor groups.

5 Must Know Facts For Your Next Test

1. If `H` is a subgroup of a group `G` and `g` is an element of `G`, then the left coset of `H` in `G` with respect to `g` is denoted by `gH` and consists of all elements of the form `gh` for `h` in `H`.
2. The number of distinct left cosets of a subgroup in a group is equal to the index of that subgroup in the group, which is represented as `[G : H]`.
3. If a subgroup `H` is normal in `G`, then all left cosets are also right cosets, meaning `gH = Hg` for all elements `g` in `G`.
4. Cosets can be used to form equivalence relations on groups, where two elements are considered equivalent if they belong to the same coset of a subgroup.
5. The collection of all left cosets of a subgroup forms a partition of the group, ensuring that every element of the group belongs to exactly one coset.

Review Questions

• How do left cosets relate to the structure and organization of groups?
  • Left cosets help illustrate how groups can be structured by showing how elements interact with subgroups. When you take an element from a group and combine it with every element from a subgroup, you create a new set that highlights how those elements work together. This helps understand concepts like partitions and equivalence relations within groups, making it easier to analyze their structure.
• Explain why understanding left cosets is essential when studying normal subgroups.
  • Understanding left cosets is essential when studying normal subgroups because normal subgroups have special properties regarding their cosets. For normal subgroups, each left coset is also a right coset, which means they behave uniformly when interacting with other elements in the group. This uniformity allows for the construction of quotient groups, which are foundational in many areas of representation theory and abstract algebra.
• Evaluate how left cosets contribute to forming quotient groups and their significance in representation theory.
  • Left cosets are integral in forming quotient groups, as they represent how a larger group can be simplified through its normal subgroups. When we take the set of all left cosets of a normal subgroup, we can construct a quotient group that retains much of the original group's structure while simplifying its interactions. This abstraction allows for deeper insights into representation theory, enabling mathematicians to study linear representations and character theory, ultimately revealing how these concepts interconnect within algebraic structures.

Related terms

Subgroup: A subgroup is a subset of a group that itself forms a group under the same operation as the original group.

Normal Subgroup: A normal subgroup is a subgroup that is invariant under conjugation by any element of the group, meaning it looks the same when viewed from any perspective within the larger group.

Group Action: Group action refers to the way a group interacts with another set, allowing elements of the group to 'act' on elements of that set in a structured manner.