Subgroup

Definition

A subgroup is a subset of a group that itself forms a group under the same operation. This means that if you take a group and pick out some of its elements, those elements must satisfy the group properties: closure, associativity, identity, and inverses. Understanding subgroups helps in studying the structure of groups and how they relate to each other through homomorphisms and other algebraic concepts.

5 Must Know Facts For Your Next Test

1. Any group always contains at least two subgroups: the trivial subgroup containing only the identity element and the group itself.
2. A subgroup must satisfy all four group properties: closure (the product of two elements in the subgroup is also in the subgroup), associativity (the operation is associative), identity (there is an identity element in the subgroup), and inverses (for every element in the subgroup, its inverse must also be in the subgroup).
3. If a subgroup is finite and non-empty, its order (number of elements) must divide the order of the parent group, as stated by Lagrange's theorem.
4. Subgroups can be used to construct factor groups, which are important in understanding quotient structures in group theory.
5. The intersection of two subgroups is also a subgroup, meaning that even when combining smaller groups, you can still find valid groups within them.

Review Questions

• How can you determine if a subset of a group qualifies as a subgroup?
  • To determine if a subset qualifies as a subgroup, you need to verify that it satisfies the four group properties: closure, associativity, identity, and inverses. First, check closure by ensuring that for any two elements in the subset, their product (or operation result) is also in the subset. Then, confirm that the associative property holds as it does for the parent group. Next, ensure that there is an identity element present in the subset. Finally, for each element in the subset, verify that its inverse also exists within that subset.
• Discuss how subgroups relate to cosets and why this relationship is important in group theory.
  • Subgroups and cosets are closely related because cosets provide a way to partition a group based on its subgroups. When you take a subgroup and multiply it by an element from the larger group, you form a coset. This allows us to explore the structure of groups more deeply. For example, understanding how many distinct cosets exist helps us determine whether or not the subgroup is normal and leads to further insights such as index calculations or constructing factor groups.
• Evaluate how Lagrange's theorem connects subgroups to the structure of groups and its implications for homomorphisms.
  • Lagrange's theorem states that the order of any subgroup must divide the order of its parent group. This theorem has significant implications for understanding group structure because it allows us to deduce possible orders of subgroups based on known quantities. In terms of homomorphisms, this relationship helps establish constraints on how groups can map into each other; for instance, if a homomorphism from one group to another exists, it impacts how subgroups correspond across those groups and can lead to insights about image and kernel sizes.

Related terms

Normal Subgroup: A normal subgroup is a subgroup that is invariant under conjugation by any element of the group, meaning it remains unchanged when elements from the group are used to combine with it.

Coset: A coset is a form of partitioning a group based on a subgroup, created by multiplying all elements of the subgroup by a fixed element from the group.

Lattice of Subgroups: The lattice of subgroups is a graphical representation showing how subgroups are related to one another in terms of inclusion, illustrating which subgroups contain others.