Index of a Subgroup

Definition

The index of a subgroup is the number of distinct left or right cosets of that subgroup in the larger group. This concept helps to understand how subgroups partition the group and plays a crucial role in various theorems and applications within group theory.

5 Must Know Facts For Your Next Test

1. The index of a subgroup can be calculated as the ratio of the order of the group to the order of the subgroup: $$ [G:H] = \frac{|G|}{|H|} $$.
2. The index is always a positive integer, and it gives insight into the structure of the group and its subgroups.
3. If a subgroup has index 1, it means that it is actually the whole group.
4. The index can also help determine properties like whether a subgroup is normal; if all left cosets equal right cosets, the subgroup is normal.
5. In finite groups, Lagrange's Theorem directly relates the index of subgroups to their orders and indicates that indices must be integers.

Review Questions

• How does the index of a subgroup relate to Lagrange's Theorem and its implications for group structure?
  • The index of a subgroup directly illustrates Lagrange's Theorem by showing that the order of a subgroup divides the order of the entire group. This means that if you know the size of the whole group and its subgroup, you can determine how many cosets exist, which is exactly what the index represents. Essentially, it provides a way to visualize and quantify the relationship between groups and their subgroups.
• Discuss how calculating the index can aid in determining whether a subgroup is normal within a given group.
  • Calculating the index can help identify whether a subgroup is normal by allowing us to analyze its left and right cosets. If these cosets coincide, indicating that they contain the same elements regardless of multiplication order with elements from the larger group, then we know that the subgroup is normal. Thus, understanding indices provides insights into subgroup properties and their role within larger groups.
• Evaluate how understanding indices can facilitate classifying groups of small order using Sylow Theorems.
  • Understanding indices becomes crucial when applying Sylow Theorems for classifying groups of small order because it helps identify potential Sylow subgroups based on their indices. By calculating indices and comparing them with factors derived from group orders, we can determine how many such subgroups exist and their respective properties. This process simplifies classification tasks by providing concrete numerical relationships among different subgroups within a small group.

Related terms

Coset:

A coset is a form of grouping in which elements are formed by multiplying a fixed element from a group by each element of a subgroup.

Lagrange's Theorem:

Lagrange's Theorem states that the order of a subgroup divides the order of the entire group, linking the sizes of groups and their subgroups.

Normal Subgroup:

A normal subgroup is a subgroup that is invariant under conjugation by elements of the group, leading to well-defined factor groups.