Subgroup

Definition

A subgroup is a subset of a group that is itself a group under the same operation as the larger group. This means that a subgroup must include the identity element, be closed under the group operation, and contain the inverse for each of its elements. Subgroups are essential in understanding the structure of algebraic groups and how they act on various spaces.

5 Must Know Facts For Your Next Test

1. Every group has at least two subgroups: the trivial subgroup (containing only the identity) and the group itself.
2. If a subgroup is normal, it allows for the formation of quotient groups, which play a significant role in understanding group actions.
3. Subgroups can be finite or infinite, depending on the number of elements they contain compared to the larger group.
4. The intersection of two subgroups is also a subgroup, meaning that subgroups can share elements while still being distinct groups.
5. The order of a subgroup divides the order of the group, according to Lagrange's theorem, linking subgroup structure directly to group size.

Review Questions

• How do you determine whether a subset is a subgroup of a given group?
  • To determine if a subset is a subgroup, you need to check three criteria: first, confirm that it contains the identity element of the original group. Second, verify that the subset is closed under the group's operation, meaning combining any two elements from the subset results in an element that is also in the subset. Lastly, check if every element in the subset has its inverse also within the subset. If all three conditions are met, then it's a subgroup.
• Discuss the significance of normal subgroups in relation to quotient groups.
  • Normal subgroups are significant because they allow for the creation of quotient groups. A normal subgroup remains unchanged when its elements are conjugated by any element of the larger group. This property enables us to define a new group where cosets of this normal subgroup form the elements. Quotient groups help us analyze more complex structures and understand how groups behave under various operations.
• Evaluate how subgroups contribute to our understanding of algebraic groups and their actions on varieties.
  • Subgroups are fundamental in studying algebraic groups because they provide insight into how these groups interact with geometric structures such as varieties. The behavior of subgroups can reveal symmetries and invariances in varieties under group actions, allowing us to classify objects based on their geometric properties. Moreover, understanding subgroups aids in analyzing representations of algebraic groups and can highlight important features such as stabilizers and orbit structures that are crucial for understanding geometric interpretations.

Related terms

Group:

A set equipped with an operation that combines any two elements to form a third element, satisfying four fundamental properties: closure, associativity, identity element, and invertibility.

Normal Subgroup:

A subgroup that is invariant under conjugation by any element of the group, meaning that for every element in the subgroup and every element in the group, their product remains within the subgroup.

Coset: A form of partitioning a group into disjoint subsets, created by multiplying all elements of a subgroup by a fixed element from the group.