Question

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

Short answer

The intersection \(H \cap K\) is normal in \(G\).

Step-by-step solution

Understand the Definitions

A subgroup \(H\) is normal in a group \(G\) (denoted \(H \triangleleft G\)) if for every element \(g \in G\), the conjugate \(gHg^{-1} = H\). Given \(H \triangleleft G\) and \(K \triangleleft G\), determine if their intersection \(H \cap K\) is also normal in \(G\).

Analyze the Intersection

The intersection \(H \cap K\) consists of all elements that are both in \(H\) and in \(K\). We need to show that for any \(g \in G\), \(g(H \cap K)g^{-1} = H \cap K\).

Verify Conjugation in Subgroups

Since \(H \triangleleft G\), we have \(gHg^{-1} = H\) for any \(g \in G\). Similarly, since \(K \triangleleft G\), we also have \(gKg^{-1} = K\). We need to show the same holds for \(H \cap K\).

Consider an Element in the Intersection

Take an element \(x \in H \cap K\). Then \(x \in H\) and \(x \in K\). By normality, \(gxg^{-1} \in H\) and \(gxg^{-1} \in K\) because \(gxg^{-1}\) is obtained by conjugation, and conjugates remain in the subgroups. Thus, \(gxg^{-1} \in H \cap K\).

Conclude the Normality of the Intersection

Since \(gxg^{-1} H \cap K\) for any \(g \in G\), this implies \(g(H \cap K)g^{-1} \subseteq H \cap K\). Also by reversing the argument, we find \((H \cap K)g^{-1} g(H \cap K)\), hence \(g(H \cap K)g^{-1} = H \cap K\). Thus, \(H \cap K\) is normal in \(G\).

Key concepts

Group Theory

Group theory is a foundational branch of mathematics that deals with groups. A group is essentially a set that is combined with an operation, like addition or multiplication, that satisfies four main properties: closure, associativity, identity, and invertibility. In group theory, the structure of the set and the operation it supports are studied.

Groups can be finite, such as the set of integers with addition, or infinite, like the real numbers under multiplication. Groups are crucial because they allow us to analyze symmetry and mathematical structures systematically.

• Closure: For any two elements \(a\) and \(b\) in the group, the result of the operation, \(a \, \ast \, b\), is also in the group.
• Associativity: The operation must satisfy \((a \, \ast \, b) \, \ast \, c = a \, \ast \, (b \, \ast \, c)\) for any elements \(a\), \(b\), \(c\) in the group.
• Identity Element: There is an element \(e\) in the group such that \(e \, \ast \, a = a \, \ast \, e = a\) for any \(a\) in the group.
• Inverse Element: For each element \(a\) in the group, there is an element \(b\) such that \(a \, \ast \, b = b \, \ast \, a = e\), where \(e\) is the identity element.

Understanding these properties helps explain why certain mathematical phenomena occur and supports other fields like algebra and geometry.

Subgroup Intersection

In the realm of group theory, a subgroup is a smaller group contained within a larger group, inheriting the structure and properties of the larger group. When we talk about the intersection of subgroups, such as \(H \cap K\), we're referring to the set that contains elements common to both subgroups \(H\) and \(K\).

This intersection also has significant properties: it is itself a subgroup. This happens because the intersection of sets preserves closure, identity, and inverses, satisfying all group requirements. When both \(H\) and \(K\) are normal subgroups—this means they are invariant under conjugation by any element of the whole group—there's an added level of structure. We can also show this intersection is normal in the parent group because individual elements follow the normality condition in both \(H\) and \(K\). This implies for any element \(g\) in the group \(G\), the conjugate of any element from the intersection by \(g\) is also in the intersection.

• For any \(g\) in \(G\), \(g(H \cap K)g^{-1} = H \cap K\).
• This preserves the essential structure and operations of the subgroups \(H\) and \(K\).

Understanding intersections enriches comprehension of subgroup dynamics within larger mathematical frameworks.

Conjugation in Groups

Conjugation in group theory is an operation where an element of the group "acts" on another element through multiplication. If \(g\) is an element of group \(G\), and \(x\) is also in \(G\), then the conjugate of \(x\) by \(g\) is given by \(gxg^{-1}\).

This operation is significant because it helps determine if a subgroup is normal. In a normal subgroup, every element remains unchanged in structure when conjugated by any element of the group. This 'unchanging' property is what characterizes normalcy in subgroups. It's a key part of the discussion about symmetric properties of groups and how they can be structured.

• A subgroup \(H\) is normal if for every \(g\) in \(G\), \(gHg^{-1} = H\).
• The concept of conjugation ensures the subgroup retains its properties under the group's operations, making it invariant under such internal permutations.
• This concept is crucial in deeper topics like group homomorphisms and quotient groups.

Conjugation not only provides insights into the group's symmetry but is also essential in proving and understanding complex group properties such as the normality of subgroups."