Problem 18 - Algebra: Pure and Applied

Question

Determine whether the indicated subgroup is normal in the indicated group. Show that if \(H \triangleleft G\) and \(K \triangleleft G\), then \(H K \triangleleft G\).

Step-by-Step Solution

Answer

Yes, if \(H \triangleleft G\) and \(K \triangleleft G\), then \(HK \triangleleft G\).

Step 1: Define Normal Subgroup Property

Recall the definition of a normal subgroup. A subgroup \(H\) of \(G\) is normal, denoted \(H \triangleleft G\), if for every element \(g\) in \(G\) and for every element \(h\) in \(H\), the element \(g^{-1}hg\) is also in \(H\). This means \(gH = Hg\) for every \(g \in G\).

Step 2: Express the Subgroup Product

Consider the product of two subgroups \(H\) and \(K\), denoted \(HK\). The set \(HK = \{hk \,|\, h \in H, k \in K\}\) contains all possible products of elements from \(H\) and \(K\).

Step 3: Verify Normality in G

To show \(HK \triangleleft G\), we must show that \(g^{-1}hkg \in HK\) for all \(g \in G\), \(h \in H\), and \(k \in K\). Given \(H \triangleleft G\), for each \(g \in G\), we have \(g^{-1}hg = h' \in H\). Similarly, since \(K \triangleleft G\), \(g^{-1}kg = k' \in K\).

Step 4: Combine Elements from H and K

Express the element \(g^{-1}hkg = (g^{-1}hg)(g^{-1}kg) = h'k'\). Since \(h' \in H\) and \(k' \in K\), the product \(h'k' \in HK\) by definition of \(HK\) being the product set of \(H\) and \(K\). Therefore, \(HK\) contains all conjugates \(g^{-1}hkg\) for \(h \in H\) and \(k \in K\).

Step 5: Conclusion

Since every conjugate of an element in \(HK\) stays within \(HK\), and since this holds for all elements in \(G\), the subgroup \(HK\) is normal in \(G\). Thus, \(HK \triangleleft G\).

Key Concepts

Group Theory

Group theory is a fundamental area of abstract algebra. It is the study of algebraic structures known as groups. In simple terms, a group is a set equipped with an operation that satisfies certain conditions.
Let's dive into those conditions:

• Closure: Performing the operation on any two elements of the group results in another element in the same group.
• Associativity: The operation must be associative, which means for any three elements, doing the operation on the first two and then the result with the third is the same as doing it on the last two and then the result with the first.
• Identity Element: There must be an element in the group that, when used in the operation with any element of the group, leaves that element unchanged.
• Inverse Element: Every element must have an inverse in the group, such that the operation of an element and its inverse gives the identity element.

These conditions help us to analyze systems and structures in mathematics and even in fields like physics and chemistry!

Subgroup Product

When dealing with groups, it is common to look at smaller subsets called subgroups. Subgroups are themselves groups under the same operation. The product of two subgroups, denoted as \( HK \), combines their elements in a specific way.
This method takes every element \( h \) from subgroup \( H \) and every element \( k \) from subgroup \( K \), and forms the set:

• \( HK = \{hk \mid h \in H, k \in K\} \)

This combined set, \( HK \), contains all possible outcomes from multiplying elements of \( H \) and \( K \) in sequence, and forms a new subgroup of the original group if \( H \) and \( K \) themselves are normal in the larger group. Understanding this multiplication forms a fundamental part of examining how subgroups interact with each other.

Normality Condition

The normality condition is a special property in group theory. A subgroup \( H \) is normal in \( G \), denoted \( H \triangleleft G \), if it is invariant under conjugation by any element of \( G \).
Here's what it means practically:

• For each element \( g \) in \( G \) and every element \( h \) in \( H \), the conjugate \( g^{-1}hg \) must also be in \( H \).

This regularity condition implies that the structure of \( H \) aligns symmetrically within the larger group \( G \). This property simplifies how elements can be shuffled around inside the group, facilitating further analysis.
Remember, a normal subgroup does not shift position under conjugation, maintaining its "shape" within the encompassing group.

Conjugate Elements

Understanding conjugate elements can initially feel tricky, but let's break it down. In the realm of group theory, a conjugate of an element \( h \) with respect to another element \( g \) is the element \( g^{-1}hg \).
This technique of conjugation demonstrates how an element can be transformed within the group setting.

• Conjugate elements bear a symmetrical relationship within groups, showing how structure and symmetry in group operations work.
• Conjugation is key for understanding the alignment of subgroups under various transformations.

Using conjugates helps establish whether a subgroup can be considered normal, as it reflects how elements integrate or "fit" within a larger group structure.
Ultimately, conjugation allows us to measure the internal symmetry of the elements within a group.