Question

Show that the intersection of two subgroups is a subgroup.

Short answer

Therefore, the intersection of two subgroups is indeed a subgroup, as it satisfies all the essential conditions of identity, closure and invertibility.

Step-by-step solution

State the Intersection

Consider two subgroups of a group, denote them as \( H \) and \( K \). The intersection of \( H \) and \( K \), denoted as \( H \cap K \), contains all elements that are common to both \( H \) and \( K \).

Demonstrate the Identity Property

An identity element exists in both the subgroups \( H \) and \( K \). This is because all groups have an identity element. Thus, the identity element also exists in the intersection of \( H \) and \( K \), that is, \( H \cap K \).

Demonstrate the Closure Property

Let's take any two elements, say \( a \) and \( b \) from the intersection \( H \cap K \). These elements will certainly also belong to both \( H \) and \( K \) since they are in the intersection. Since \( H \) and \( K \) are subgroups, the product \( a \cdot b \) or the operation between \( a \) and \( b \) must also belong to \( H \) and \( K \). Consequently, \( a \cdot b \) must also belong to the intersection \( H \cap K \), i.e., closure property is also verified.

Demonstrate the Invertibility Property

For an element \( a \) in the intersection \( H \cap K \), it is also in both subgroups \( H \) and \( K \) since it is part of their intersection. Since \( H \) and \( K \) are subgroups, the inverse of \( a \), denoted by \( a^{-1} \), also belongs to \( H \) and \( K \). That means \( a^{-1} \) also belongs to the intersection \( H \cap K \). This verifies the invertibility property.

Conclusion

Since the intersection \( H \cap K \) satisfies the three necessary properties of identity, closure, and invertibility, it can be concluded that the intersection of any two subgroups \( H \) and \( K \) of a group is also a subgroup of the same group.

Key concepts

Subgroup

In Group Theory, a subgroup is essentially a smaller group within a larger group. To be a subgroup, a set must adhere to three basic properties:

• It must contain the identity element of the larger group.
• It must be closed under the group operation, meaning the result of the operation on two elements from the subgroup remains in the subgroup.
• Every element in the subgroup must have an inverse that is also within the subgroup.

These properties ensure that subgroups share similar characteristics to the main group, allowing them to function independently while still being a part of the whole. When solving problems involving subgroups, always check for these properties.

Intersection

The intersection of sets is the collection of elements that are common to all sets in question. In the context of group theory, the intersection of two subgroups, say \( H \) and \( K \), is denoted by \( H \cap K \). This intersection contains all elements that are shared by both \( H \) and \( K \). To determine if \( H \cap K \) is itself a subgroup, you must verify it satisfies the subgroup properties.

• It must have the identity element.
• It should be closed under its operation.
• It must contain inverses of all its elements.

By fulfilling these conditions, \( H \cap K \) is confirmed as a subgroup of the larger group.

Identity Element

In any group, the identity element is a special member that leaves other elements unchanged when used in the group operation. For a group \( G \) with operation \( * \), if \( e \) is the identity element, then for any element \( a \) in \( G \), we have \( e * a = a * e = a \).
In subgroup theory, the identity element of the main group should also be present in any subgroup, including intersections of subgroups, like \( H \cap K \). Having this element is crucial for maintaining the group's defining properties. Think of the identity element as a neutral starting point within the subgroup.

Closure Property

The closure property deals with performing operations within a subgroup. If \( a \) and \( b \) are elements of a subgroup \( H \), then the result of the operation, \( a \cdot b \), should still be in \( H \). Likewise, if \( H' \) and \( K' \) are intersecting subgroups, their intersection \( H' \cap K' \) should also be closed.

• This means the operation of any two elements in the intersection must yield another element that is in the intersection.
• It is a fundamental aspect of subgroup criteria, ensuring consistency under the group operation.

The closure property maintains the subgroup's structural integrity within the larger group. Without closure, the subgroup cannot function independently within the group.

Invertibility Property

The invertibility property of a group asserts that for every element \( a \) in a subgroup, there exists an inverse \( a^{-1} \) such that the operation on \( a \) and \( a^{-1} \) returns the identity element of the group. This is crucial for maintaining balance within the group structure.

• In the intersection of two subgroups \( H \) and \( K \), every element in \( H \cap K \) must have an inverse in \( H \cap K \).
• This property guarantees that all operations can be undone, a core feature of group operations.

Verifying invertibility helps ensure that the intersection is indeed a subgroup. This step is essential, as it ensures all the subgroup's elements can "go back" to their identity state.