Problem 25 - Algebra: Pure and Applied

Question

Let \(G\) be a group and \(S\) a subset of \(G\). Define \(\langle S\rangle\) to be the smallest subgroup of \(G\) containing \(S\), called the subgroup of \(G\) generated by \(S .\) Show that \(\langle S\rangle\) exists.

Step-by-Step Solution

Answer

The subgroup \( \langle S \rangle \) exists as the intersection of all subgroups of \( G \) containing \( S \).

Step 1: Use the Concept of Subset Intersection

The subgroup generated by a subset \( S \) of \( G \), \( \langle S \rangle \), can be defined as the intersection of all subgroups of \( G \) that contain \( S \). If each subgroup contains \( S \), their intersection must also contain \( S \).

Step 2: Show Existence of Subgroup Containing S

Consider the entire group \( G \) itself as one of the subgroups containing \( S \). The set of all subgroups of \( G \) that contain \( S \) is non-empty since \( G \) itself is included. Therefore, this intersection is a well-defined non-empty set.

Step 3: Verify Subgroup Properties

The intersection of subgroups is again a subgroup because it satisfies the group properties: closure, identity, and inverses. Specifically, if \( H_\alpha \) are subgroups containing \( S \), then their intersection \( H = \bigcap_\alpha H_\alpha \) also satisfies: - If \( a, b \in H \), then \( a b^{-1} \in H \). - The identity element exists in each \( H_\alpha \), hence in \( H \). - For each \( a \in H \), the inverse \( a^{-1} \in H \).

Step 4: Establish Minimality of the Subgroup

The group \( \langle S \rangle = \bigcap_{\alpha} H_{\alpha} \) is the smallest subgroup containing \( S \) by construction since any subgroup of \( G \) that contains \( S \) must include every element of this intersection. Therefore, no smaller subgroup containing \( S \) can exist.

Step 5: Conclude Existence and Uniqueness

Since \( \langle S \rangle = \bigcap_{\alpha} H_{\alpha} \), which is a subgroup and the smallest by construction, \( \langle S \rangle \) exists uniquely for any subset \( S \). Thus the subgroup generated by \( S \) always exists.

Key Concepts

Understanding Subgroups

In the study of group theory, a subgroup is a group formed from a larger group that inherits the properties of the parent group. To be a subgroup, a subset of a group must satisfy specific criteria:

• Closure: For any two elements in the subset, their product must also be in the subset.
• Identity: There is an identity element in the subset that satisfies the identity property of the group.
• Inverses: For every element in the subset, there is an inverse element that is also in the subset.

These conditions ensure that the subset behaves like a group within the larger group. Subgroups are important in understanding the structure of groups since examining these smaller assemblies can shed light on the properties of the larger group itself.

Examining Group Properties

Group properties are foundational to understanding how groups, including subgroups, operate. A group is a set paired with an operation that satisfies certain criteria:

• Associativity: The operation on any three elements of the set is associative.
• Identity: There is an element in the group that leaves any element unchanged when combined with it.
• Inverse: Every element has an inverse such that when the element and its inverse are combined, the result is the identity.

When we talk about subgroups, it's crucial to recognize that they also obey these group properties. This means any subgroup of a group is itself a group, because it must satisfy associativity, possess an identity element, and ensure each element has an inverse. Understanding these properties allows us to predict and leverage the behaviors of groups and their subgroups.

Intersection of Subgroups

The concept of intersecting subgroups is vital when analyzing group structures. If you have multiple subgroups within a parent group that all contain a certain subset, their intersection will also be a subgroup. This is because:

• Closure is maintained, as any combination of elements that is valid in each subgroup remains valid in the intersection.
• The identity element exists in each subgroup, ensuring it exists in the intersection.
• Inverses of all elements are guaranteed, since they appear in every intersected subgroup.

This means that the intersection of subgroups not only still behaves as a group, but is also the most common set of elements that ties all participating subgroups together. It is minimal, yet highly structured, which is a useful property when attempting to distill the essence of a group's composite structure.

Generated Subgroup

A generated subgroup is a uniquely important idea in group theory. Given a subset of a group, the subgroup generated by that subset, denoted as \( \langle S \rangle \), is the smallest subgroup containing that subset. To construct it:

• Consider every subgroup of the larger group that contains this subset.
• The intersection of all such subgroups is \( \langle S \rangle \).

This generated subgroup is uniquely defined and is essential because it provides a structured way to encapsulate all elements that can be created using the subset under the group's operation. Its existence is guaranteed, as discussed in the intersection of subgroups, since intersecting over all subgroups containing the subset will still satisfy the subgroup properties. This makes \( \langle S \rangle \) a key tool for understanding how groups can be built from smaller pieces in a comprehensive and orderly fashion.