Problem 26 - Algebra: Pure and Applied

Question

Let \(H\) and \(K\) be subgroups of a group \(G\). Show that \(H K\) is a subgroup of \(G\) if and only if \(H K=K H\).

Step-by-Step Solution

Answer

\( HK \) is a subgroup of \( G \) if and only if \( HK = KH \).

Step 1: Understanding the Problem

We need to show that the product of two subgroups \( H \) and \( K \), denoted as \( HK \), is also a subgroup of \( G \) if and only if \( HK = KH \). This means we need to show two implications: (1) If \( HK \) is a subgroup, then \( HK = KH \). (2) If \( HK = KH \), then \( HK \) is a subgroup.

Step 2: Proving First Implication

Assume \( HK \) is a subgroup of \( G \). For any \( h \in H \) and \( k \in K \), \( hk \in HK \). Since \( HK \) is a subgroup, it is closed under taking inverses. Hence, \((hk)^{-1} = k^{-1}h^{-1} \in HK \). This implies there exist \( h' \in H \) and \( k' \in K \) such that \( k^{-1}h^{-1} = h'k' \). Rearranging gives \( hk = k'(h')^{-1} \), showing that each element of \( HK \) can be expressed as any element of \( KH \), hence \( HK \subseteq KH \). By symmetry, \( KH \subseteq HK \), thus \( HK = KH \).

Step 3: Proving Second Implication

Assume \( HK = KH \). We need to show that \( HK \) is a subgroup. To demonstrate subgroup properties, we need to show closure and inverses are contained in \( HK \). For closure, take any \( hk, h'k' \in HK \). Since \( KH = HK \), reorder and combine: \( hk \cdot h'k' = h(hk'h')k' \in HK \). For inverses, \( (hk)^{-1} = k^{-1}h^{-1} \). Since \( KH = HK \), \( k^{-1}h^{-1} \) can be expressed as an element of \( KH \), and thus in \( HK \), proving that \( HK \) is a subgroup.

Step 4: Conclusion

We have shown both implications: If \( HK \) is a subgroup then \( HK = KH \), and if \( HK = KH \) then \( HK \) is a subgroup. Therefore, \( HK \) is a subgroup of \( G \) if and only if \( HK = KH \).

Key Concepts

Group Operation

In group theory, a **group operation** is a fundamental concept that defines how elements combine within a group. A group is a set equipped with an operation that takes any two elements of the group and produces another element of the group.

• **Associativity**: The operation must satisfy the associative law, meaning that for any elements \( a, b, \) and \( c \) in the group, \( (ab)c = a(bc) \).
• **Identity Element**: There exists an identity element \( e \) in the group such that for any element \( a \), \( ae = ea = a \).
• **Inverse Element**: For every element \( a \) in the group, there exists an inverse element \( a^{-1} \) such that \( aa^{-1} = a^{-1}a = e \).

To visualize a group operation, think of it like a set of rules that govern how elements "interact" inside the group. It's what conjoins the abstract concept of a group with tangible actions like adding, multiplying, or even more complex operations. Understanding this operation is key when studying cosets, as it helps explain how subgroups mesh together to form new structures within a larger group.

Cosets

Cosets are collections derived from a subgroup within a larger group. They are used to partition groups into disjoint subsets. Given a subgroup \( H \) of a group \( G \), any element of \( G \) can be paired with \( H \) to form what is called a **coset**.

• **Left Coset**: For an element \( g \) in \( G \), the left coset of \( H \) in \( G \) is \( gH = \{ gh : h \in H \} \).
• **Right Coset**: Similarly, the right coset of \( H \) with respect to \( g \) is \( Hg = \{ hg : h \in H \} \).

The nice thing about cosets is that they either completely overlap or are completely disjoint. This means that they partition the group \( G \) neatly without any gaps. Cosets become very important when analyzing group structures. They help in building quotient groups and analyzing how subgroups interact with one another through the group operation. In subgroup research, knowing whether cosets are equivalent depends significantly on the subgroup properties and helps in proving statements such as \( HK = KH \).

Closure Property

The **closure property** is a vital component of group theory that ensures for any elements \( a \) and \( b \) in a group, the result of the group operation, \( ab \), is also in the group. This property is a required condition for any subset to be considered a group.When dealing with sets like \( HK \) (the product of subgroups \( H \) and \( K \) of a group \( G \)), closure ensures that every possible combination of elements from \( H \) and \( K \) stays within the group \( G \).

• **Element Pairing**: If \( h \in H \) and \( k \in K \), then the product \( hk \) must be in \( HK \) if it is a group.
• **Subgroup Combination**: The operation must behave such that any combination \( hk \) and further products \( (hk)(h'k') \) remain in \( HK \), verifying the closure law for \( HK \).

Applying closure is straightforward but supremely important. Without closure, the set falls apart, breaking the consistency expected in group operations. In essence, closure maintains the integrity of a group's structure, ensuring that subgroup operations like those seen in \( H \) and \( K \) produce valid results within the larger group \( G \) and can be repeated indefinitely.