Left coset

Definition

A left coset of a subgroup H in a group G is defined as the set of all elements obtained by multiplying a fixed element g from G by each element h in H. This is expressed as gH = {gh | h ∈ H}. Left cosets help us understand the structure of groups by partitioning them into equivalent classes based on the subgroup's elements, revealing important properties related to normal subgroups and quotient groups.

5 Must Know Facts For Your Next Test

1. Left cosets can be used to partition a group into disjoint sets, meaning any two different left cosets do not share any elements.
2. If H is a finite subgroup of G, then all left cosets of H in G have the same number of elements as H.
3. The left coset gH is equal to the left coset g'H if and only if g and g' are in the same left coset, which relates to the concept of equivalence classes.
4. If H is a normal subgroup of G, then all left cosets are also right cosets, leading to simpler structures in quotient groups.
5. The index of H in G, denoted as [G:H], is the number of distinct left cosets of H in G, providing insight into the relationship between subgroup sizes and group structure.

Review Questions

• How does the concept of left cosets relate to understanding the structure of groups and their subgroups?
  • Left cosets allow us to break down a group into manageable pieces based on its subgroups. By forming left cosets from a subgroup, we can see how elements of the larger group relate to one another. This partitioning helps in identifying properties such as whether certain subgroups are normal or how they interact with other elements in the group, thus shedding light on the overall structure.
• Discuss how left cosets can help determine if a subgroup is normal within its parent group.
  • To determine if a subgroup H is normal in a group G, we can analyze its left and right cosets. If every left coset gH equals its corresponding right coset Hg for all g in G, this shows that H is invariant under conjugation by elements of G. This equality indicates that H is normal, allowing us to construct quotient groups that reflect this symmetry.
• Evaluate the importance of left cosets in the context of forming quotient groups and understanding their properties.
  • Left cosets are fundamental when forming quotient groups because they create equivalence classes based on the subgroup's structure. When a subgroup is normal, the distinct left cosets can be combined to create a new group, known as a quotient group. This new group's structure reflects how elements interact with one another modulo the normal subgroup, enabling deeper insights into both algebraic and geometric properties within group theory.

Related terms

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

Normal subgroup: A normal subgroup is a subgroup that is invariant under conjugation by any element of the group, meaning it can be used to form quotient groups.

Quotient group: A quotient group is formed by partitioning a group into cosets of a normal subgroup, which allows for the study of group structures at a higher level.