Understanding the Normalizer in Group Theory
In group theory, the Normalizer is a fundamental concept that helps us understand how elements and subgroups interact within a group. This lesson provides clear definitions, practical examples, and detailed proofs of important theorems. By exploring concepts such as the centralizer (often regarded as a type of normalizer) and conjugate elements, you will build a solid foundation in understanding normal subgroups and their role in abstract algebra.
Firstly, we discuss the centralizer or normalizer of an element. Then, we explore the idea of conjugate elements, which naturally leads to important theorems. Consequently, these theorems illustrate the properties and structure of the normalizer and its significance in group theory. Throughout the lesson, transitional phrases like “furthermore” and “for instance” will guide you through each logical step.
Centralizer (Normalizer) of an Element
Definition:
Let \((G, \circ)\) be a group and \(a \in G\). The centralizer (or normalizer in this context) of \(a\) in \(G\) is defined by
\[
C(a) = \{ x \in G : a \circ x = x \circ a \}.
\]
In other words, \(C(a)\) consists of all elements in \(G\) that commute with \(a\). This set is crucial for understanding the symmetry and structure within groups, particularly when analyzing normal subgroups.
Conjugate of an Element
Definition:
For any element \(a \in G\) of a group \((G, \circ)\), an element \(b \in G\) is called a conjugate of \(a\) if there exists some \(x \in G\) such that
\[
b = x \circ a \circ x^{-1}.
\]
This relationship shows how the normalizer operates through conjugation. For instance, in the symmetric group \(S_3\), conjugates of a 2-cycle yield other 2-cycles, demonstrating the inherent symmetry within the group.
Theorem 1: Center Elements and Equality of the Centralizer
Statement:
Let \((G, \circ)\) be a group and \(a \in G\). If \(a\) belongs to the center \(Z(G)\) of \(G\), then the centralizer \(C(a)\) is equal to the entire group \(G\).
Proof:
-
Showing \(C(a) \subseteq G\):
For any \(x \in C(a)\), by definition, \(x \in G\). Hence, \(C(a) \subseteq G\). -
Showing \(G \subseteq C(a)\):
Let \(y \in G\). Since \(a \in Z(G)\), we have \(a \circ y = y \circ a\). Therefore, every \(y\) in \(G\) commutes with \(a\), implying \(y \in C(a)\). Consequently, \(G \subseteq C(a)\).
Combining both inclusions, we conclude that \(C(a) = G\). This theorem emphasizes that when an element lies in the center, its normalizer naturally encompasses the whole group.
Theorem 2: The Centralizer is a Subgroup
Statement:
Let \((G, \circ)\) be a group and \(a \in G\). Then the centralizer \(C(a)\) forms a subgroup of \(G\).
Proof:
-
Non-emptiness:
The identity element \(e\) of \(G\) satisfies \(e \circ a = a \circ e\), so \(e \in C(a)\). Thus, \(C(a)\) is not empty. -
Closure under the group operation:
Assume \(x, y \in C(a)\). Then, \[ x \circ a = a \circ x \quad \text{and} \quad y \circ a = a \circ y. \] Therefore, \[ (x \circ y) \circ a = x \circ (y \circ a) = x \circ (a \circ y) = (x \circ a) \circ y = (a \circ x) \circ y = a \circ (x \circ y). \] Hence, \(x \circ y \in C(a)\). -
Existence of inverses:
For any \(x \in C(a)\), we have \(x \circ a = a \circ x\). Multiplying both sides by \(x^{-1}\) appropriately shows that \[ a \circ x^{-1} = x^{-1} \circ a, \] meaning \(x^{-1} \in C(a)\).
Therefore, \(C(a)\) satisfies all subgroup criteria and is indeed a subgroup of \(G\). This property of the normalizer is essential for deeper structural analysis in group theory.
Theorem 3: Conjugacy as an Equivalence Relation
Statement:
Let \((G, \circ)\) be a group. Define the relation
\[
\rho = \{ (x, y) \in G \times G : y \text{ is a conjugate of } x \}.
\]
We claim that \(\rho\) is an equivalence relation.
Proof:
-
Reflexivity:
For any \(a \in G\), using the identity element \(e\) we have: \[ a = e \circ a \circ e^{-1}. \] Hence, \((a, a) \in \rho\). -
Symmetry:
Assume \((a, b) \in \rho\). Then, \(b = x \circ a \circ x^{-1}\) for some \(x \in G\). It follows that: \[ a = x^{-1} \circ b \circ x, \] so \((b, a) \in \rho\). -
Transitivity:
Suppose \((a, b) \in \rho\) and \((b, c) \in \rho\), meaning there exist \(x, y \in G\) such that: \[ b = x \circ a \circ x^{-1} \quad \text{and} \quad c = y \circ b \circ y^{-1}. \] Then, \[ c = y \circ (x \circ a \circ x^{-1}) \circ y^{-1} = (y \circ x) \circ a \circ (y \circ x)^{-1}. \] Therefore, \((a, c) \in \rho\).
By establishing reflexivity, symmetry, and transitivity, we conclude that \(\rho\) is indeed an equivalence relation. This result further demonstrates how conjugation partitions a group into classes—a key concept related to the normalizer.
Related Docs
FAQs
Group Theory
- What is a group in group theory?
A group is a set equipped with a binary operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverses for every element.
- What are the main properties of a group?
The four main properties are:
- Closure: The result of the operation on any two elements of the group is also in the group.
- Associativity: The group operation is associative.
- Identity: There exists an element that does not change other elements when used in the operation.
- Invertibility: Every element has an inverse that, when combined with the element, yields the identity.
- What is the identity element in a group?
The identity element is a unique element in the group that, when combined with any other element using the group operation, leaves that element unchanged. It is commonly denoted by e or 1.
- What is an abelian group?
An abelian group is one in which the binary operation is commutative. This means for any two elements a and b in the group, a · b = b · a
- What is a subgroup?
A subgroup is a subset of a group that is itself a group under the same binary operation. It must satisfy the group properties: closure, associativity, identity, and inverses.
- What is a normal subgroup and how does it relate to quotient groups?
A normal subgroup is a subgroup that is invariant under conjugation by any element of the original group. This means for every element n in the normal subgroup N and every element g in the group, gng⁻¹ is still in N. Normal subgroups allow the construction of quotient groups, where the group is partitioned into cosets of the normal subgroup.
- What are group homomorphisms?
A group homomorphism is a function between two groups that preserves the group operation. This means if f: G → H is a homomorphism and a, b are elements of G, then f(a · b) = f(a) · f(b) in H.
- What is Lagrange’s theorem in group theory?
Lagrange's theorem states that for any finite group, the order (number of elements) of every subgroup divides the order of the entire group. This theorem is a fundamental result in the study of finite groups.
- What is Cayley’s theorem?
Cayley’s theorem states that every group is isomorphic to a subgroup of a symmetric group. This implies that every group can be represented as a group of permutations acting on a set.
- How is group theory applied in other fields?
Group theory has applications in many fields including:
- Physics: Describing symmetries and conservation laws.
- Chemistry: Analyzing molecular symmetry and chemical bonding.
- Cryptography: Underlying structures in cryptographic systems.
- Mathematics: Foundational in algebra, geometry, and number theory.
