What Is A Normal Subgroup

Delving Deep into Normal Subgroups: A Comprehensive Guide

Understanding normal subgroups is crucial for anyone studying group theory, a fundamental branch of abstract algebra. This comprehensive guide will explore what normal subgroups are, why they're important, and how to identify them. We'll delve into the core concepts, providing clear explanations and examples to solidify your understanding. By the end, you'll have a solid grasp of this essential algebraic structure and its significance in various mathematical applications.

Introduction: What are Groups and Subgroups?

Before diving into normal subgroups, let's refresh our understanding of groups and subgroups. A group is a set G equipped with a binary operation (often denoted by · or +) that satisfies four axioms:

Closure: For all a, b in G, a · b is also in G.
Associativity: For all a, b, c in G, (a · b) · c = a · (b · c).
Identity Element: There exists an element e in G (called the identity element) such that for all a in G, a · e = e · a = a.
Inverse Element: For every a in G, there exists an element a⁻¹ in G (called the inverse of a) such that a · a⁻¹ = a⁻¹ · a = e.

A subgroup H of a group G is a subset of G that is itself a group under the same operation as G. This means H must satisfy the four group axioms. Crucially, the identity element of H must be the same as the identity element of G, and the inverse of any element in H must also be in H.

Understanding Normal Subgroups: The Defining Property

A normal subgroup, often denoted as N ⊲ G, is a special type of subgroup that possesses a key property relating to its interaction with the group G. This defining characteristic involves conjugation.

The conjugate of an element h in a subgroup H by an element g in G is defined as g⁻¹hg. A subgroup H is a normal subgroup of G if and only if for every element h in H and every element g in G, the conjugate g⁻¹hg is also in H. In simpler terms: every conjugate of every element in H is still within H.

This seemingly simple condition has profound consequences for the structure and properties of the group.

Why are Normal Subgroups Important?

The significance of normal subgroups stems from their role in forming quotient groups (also known as factor groups). A quotient group is a new group constructed from a group G and one of its normal subgroups N. The elements of the quotient group G/N are the cosets of N in G.

A left coset of N in G is a set of the form gN = {gn | n ∈ N}, where g is an element of G. Similarly, a right coset is Ng = {ng | n ∈ N}. Crucially, if N is a normal subgroup, then the left and right cosets are identical (gN = Ng for all g in G). This equality is the key to defining the group operation on the cosets. The operation on cosets is defined as (g₁N)(g₂N) = (g₁g₂)N. This operation is well-defined only if N is a normal subgroup.

The ability to form quotient groups unlocks powerful tools in group theory, allowing for the analysis of group structure by considering simpler, "smaller" groups. This is fundamental to understanding the structure of more complex groups.

How to Identify a Normal Subgroup

Several methods can be used to determine whether a subgroup is normal:

The Conjugation Test: This is the direct application of the definition. For every element h in H and every element g in G, verify that g⁻¹hg is in H. This method is straightforward but can be computationally intensive for large groups.
Index 2 Subgroups: If a subgroup H has index 2 in G (meaning there are exactly two left cosets of H in G), then H is automatically a normal subgroup. This is a convenient shortcut because it avoids the need to check conjugation for every element.
Commutative Groups: In a commutative (or Abelian) group, every subgroup is a normal subgroup. This simplifies the analysis significantly. In a commutative group, the operation is commutative: a · b = b · a for all a, b in the group. Consequently, g⁻¹hg = gg⁻¹h = h, which is always in H.

Examples Illustrating Normal Subgroups

Let's consider some concrete examples to solidify our understanding:

Example 1: The trivial subgroup

The trivial subgroup {e} (containing only the identity element) is always a normal subgroup of any group G. This is because the conjugate of the identity element is always the identity element (g⁻¹eg = e).

Example 2: The entire group

The entire group G is also always a normal subgroup of itself. This is because the conjugate of any element in G is still within G.

Example 3: Subgroups of index 2

Consider the group of symmetries of a square, denoted as D₄. This group has 8 elements. Let H be the subgroup consisting of rotations. H has index 2 in D₄. Therefore, H is a normal subgroup of D₄.

Example 4: A Non-Normal Subgroup

Consider the group of permutations of three elements, S₃. Let H be the subgroup {(1), (1 2)}. This subgroup is not normal because, for example, if we conjugate (1 2) by (1 3), we get (1 3)⁻¹(1 2)(1 3) = (2 3), which is not in H.

Normal Subgroups and Homomorphisms

Normal subgroups play a vital role in understanding homomorphisms. A homomorphism is a map φ: G → G' between two groups G and G' that preserves the group operation: φ(a · b) = φ(a) · φ(b) for all a, b in G. The kernel of a homomorphism φ, denoted as ker(φ), is the set of all elements in G that map to the identity element in G'. The kernel of any homomorphism is always a normal subgroup of G. This connection establishes a fundamental link between normal subgroups and the structural properties of groups revealed through homomorphisms. This is crucial in understanding the First Isomorphism Theorem, a cornerstone result in group theory.

Frequently Asked Questions (FAQ)

Q: What makes a normal subgroup "normal"?

A: A normal subgroup N of G satisfies the condition that for every element g in G and every element n in N, the conjugate g⁻¹ng is also in N. This means that the subgroup is invariant under conjugation by elements of the group.

Q: Is every subgroup a normal subgroup?

A: No. Many subgroups are not normal. The condition for normality is a specific requirement that not all subgroups satisfy.

Q: Why are quotient groups only defined for normal subgroups?

A: The operation on cosets in a quotient group (g₁N)(g₂N) = (g₁g₂)N is only well-defined if N is normal. If N is not normal, the resulting operation on cosets is not well-defined, meaning the outcome of the operation is not uniquely determined.

Q: What is the significance of the First Isomorphism Theorem?

A: The First Isomorphism Theorem establishes a fundamental connection between homomorphisms, normal subgroups, and quotient groups. It states that for any homomorphism φ: G → G', the quotient group G/ker(φ) is isomorphic to the image of φ (im(φ)). This theorem provides a powerful tool for understanding the structure of groups by relating them to simpler quotient groups.

Conclusion: A Cornerstone of Group Theory

Normal subgroups are a fundamental concept in group theory. Their defining property of invariance under conjugation leads to the construction of quotient groups, a powerful tool for analyzing group structure. Understanding normal subgroups is essential for grasping deeper concepts in abstract algebra, including homomorphisms, isomorphism theorems, and the broader classification of groups. While the definition might appear initially complex, a solid understanding through practice and examples will reveal their elegance and importance within the rich landscape of group theory. By mastering the concepts presented here, you will significantly enhance your comprehension of this essential area of mathematics.