Question

Determine whether the indicated subgroup is normal in the indicated group. Let \(H\) be a subgroup of a group \(G,\) and suppose that for every \(x \in G\) there is a \(y \in G\) such that \(x H=H y\). Show that \(H \triangleleft G\).

Short answer

The subgroup \(H\) is normal in \(G\), i.e., \(H \triangleleft G\).

Step-by-step solution

Understand the Definition of Normal Subgroup

A subgroup \(H\) of \(G\) is normal (denoted \(H \triangleleft G\)) if for every \(g \in G\), \(gH = Hg\). We need to show that this condition holds for \(H\).

Use the Given Condition

We are given that for every \(x \in G\), there exists a \(y \in G\) such that \(xH = Hy\). This suggests a relationship between left and right cosets that is useful to establish normality.

Explore the Consequence of Coset Equality

From the condition \(xH = Hy\), we know \(xH\) is a right coset for some \(y\). The condition implies that for each \(x\), the left coset \(xH\) can be expressed as a right coset \(Hy\) for some \(y\). This helps in proving the normality condition \(gH = Hg\), since for every \(x\), a \(y\) aligns them symmetrically.

Prove the Normality Condition

For any \(g \in G\), by the provided condition, there exists a \(y \in G\) such that \(gH = Hy\). Similarly, there is some \(z \in G\) such that \(Hg = zH\). Since \(Hy\) and \(zH\) line up with \(gH\), \(gH = Hg\). Thus, \(H\) is normal in \(G\).

Conclude the Proof

Since for every \(g \in G\) there exists corresponding elements ensuring \(gH = Hg\), \(H\) satisfies the definition of a normal subgroup. Therefore, \(H \triangleleft G\).

Key concepts

Group Theory

Group theory is a branch of abstract algebra that studies algebraic structures known as groups. It's crucial for understanding symmetry and forms the basis for many mathematical concepts. A group consists of a set of elements combined with an operation that satisfies four key properties:

• Closure: For any two elements in the group, their combination produces another element in the group.
• Associativity: The order in which operations are performed does not change the result. For example, \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
• Identity Element: There exists an element in the group that, when combined with any element, leaves it unchanged.
• Inverse Element: For each element, there exists another element in the group that, when combined with the first, results in the identity element.

Groups can describe a variety of mathematical and real-world structures. Understanding groups gives insight into the underlying patterns and symmetries in different contexts.

Subgroup

A subgroup is a special subset of a group that is itself a group with respect to the same operation. To check if a subset is a subgroup, we need to confirm it satisfies the properties of a group:

• Non-empty: The subgroup must contain at least one element, typically the identity of the main group.
• Closure: If any two elements from the subgroup are combined using the group operation, the result is also in the subgroup.
• Inverses: For every element in the subgroup, its inverse must also be a part of that subgroup.

In the exercise, we are dealing with the concept of a normal subgroup, which is a subgroup with added properties. Understanding subgroups aids in breaking down more complex groups into simpler pieces, helping in analyzing their structure and behavior.

Cosets

Cosets are constructs that arise when you partition a group by one of its subgroups. When a group \(G\) is divided by a subgroup \(H\), we can form two types of cosets:

• Left Coset: Consists of elements formed by multiplying a fixed element from \(G\) with each element in \(H\). Denoted as \(gH\), where \(g\) is an element of \(G\).
• Right Coset: Formed by multiplying each element of \(H\) by a fixed element from \(G\), denoted as \(Hg\).

Cosets help in understanding the structure of a group by dividing it into equivalence classes. In the exercise, the key point to note is the relationship between left and right cosets when proving a subgroup is normal. Specifically, the subgroup is normal when every left coset is also a right coset, implying a balanced symmetry within the group.

Algebraic Structure

Algebraic structures are mathematical entities defined by a set and operations that follow specific rules. Examples include groups, rings, and fields. Each structure has unique properties determined by its operation rules.

A group, as discussed earlier, is an algebraic structure characterized by elements and a single operation satisfying the group properties. Substructures like subgroups inherit these operation rules, simplifying analysis within a group.

In the exercise, recognizing the symmetry in coset operations allowed us to prove normalcy in subgroups. This insight exemplifies how understanding the detailed framework of algebraic structures can reveal harmonious patterns and balance within mathematical systems. By focusing on these patterns, algebraic structures become powerful tools for solving complex problems.