Normalizer and Conjugate of an Element

Normalizer and Conjugate are fundamental concepts in Mathematics and Abstract Algebra. These notions provide insights into the structure of groups and their elements, introduced in the 19th century to simplify group-related problems. They are widely studied in Group Theory due to their relevance in symmetry and algebraic structures.

What You Will Learn?

In this post, you will explore:

Definition: Normalizer of an Element
Definition: Conjugate of an Element
Theorem-1: Let $(G,\circ)$ be a group and $a\in G$ . If $a\in Z(G)$ then $C(a)=G$ .
Theorem-2: Let $(G,\circ)$ be a group and $a\in G$ . Then $C(a)$ is subgroup of $G$ .
Theorem-3: Let $(G,\circ)$ be a group. Prove that the relation $\begin{align*} \rho=\set{(x,y)\in G\times G: y \text{ is a conjugate of } x} \end{align*}$ is an equivalence relation.

Things to Remember

Before diving into this post, make sure you are familiar with: Basic Definitions and Concepts of

Set Theory
Relations
Mappings
Group Theory

Introduction

Normalizer and Conjugate are significant topics in Abstract Algebra. The normalizer of a subset provides the largest subgroup where the subset remains invariant under conjugation. Similarly, the conjugate of an element demonstrates symmetry and equivalence within group structures. These concepts are crucial for solving Abstract Algebra Questions and understanding advanced algebraic frameworks.

Normalizer of an Element

Definition:
Let $(G,\circ)$ be a group and $a\in G$ . Then the centralizer or normalizer of $a$ in $G$ , is denoted by $C(a)$ and defined by $\begin{align*} C(a)=\set{x\in G: a\circ x= x\circ a} \end{align*}$

Conjugate of an Element

Definition:
Let $(G,\circ)$ be a group and $a\in G$ . Then the conjugate $b\in G$ of $a$ is defined by $b=x\circ a\circ x^{-1}$ for some $x\in G$ .

Theorem-1

Statement:
Let $(G,\circ)$ be a group and $a\in G$ . If $a\in Z(G)$ then $C(a)=G$ .

  Proof:
  Given that $(G,\circ)$ is a group and $a\in G$ .
  Let $a\in Z(G)$
  To prove $C(a)=G$

To prove $C(a)\subseteq G$
Let $\begin{align*} & x\in C(a) \\ \implies & x\in G \end{align*}$ $\therefore C(a)\subseteq G$ .
To prove $G \subseteq C(a)$
Let $\begin{align*} & y\in G \\ \implies & a\circ y=y\circ a~\big[\because a\in Z(G) \big] \\ \implies & y\in C(a) \end{align*}$ $\therefore G \subseteq C(a)$ .

Hence $C(a)=G$ .

Theorem-2

Statement:
Let $(G,\circ)$ be a group and $a\in G$ . Then $C(a)$ is subgroup of $G$ .

  Proof:
  Given that $(G,\circ)$ is a group and $a\in G$ .
  To prove $C(a)$ is subgroup of $G$

To prove $C(a)\ne \phi$
Let $e$ is the identity element of $G$ then $\begin{align*} & e \circ a= a \circ e \\ \implies & e \in C(a)\\ \implies & C(a)\ne \phi \end{align*}$
To prove $x\circ y \in C(a)~\forall~x,y\in C(a)$
Let $x,y\in C(a)$ then we have $\begin{align*} x \circ a= a \circ x \text{ and } y \circ a= a \circ y \end{align*}$ Now $\begin{align*} \big( x\circ y\big) \circ a &= x\circ \big(y \circ a\big)\\ &= x\circ \big(a \circ y\big)\\ & = \big( x\circ a\big) \circ y\\ & = \big( a\circ x\big) \circ y\\ & = a\circ \big(x \circ y\big)\\ \end{align*}$ $\therefore x\circ y \in C(a)$ .
To prove $x^{-1} \in C(a)~\forall~x\in C(a)$
Let $x\in C(a)$ then we have $\begin{align*} & x \circ a= a \circ x \\ \implies & a\circ x^{-1}= x^{-1} \circ a \\ \end{align*}$ $\therefore x^{-1} \in C(a)$ .

Hence $C(a)$ is subgroup of $G$ .

Theorem-3

Statement:
Let $(G,\circ)$ be a group. Prove that the relation $\begin{align*} \rho=\set{(x,y)\in G\times G: y \text{ is a conjugate of } x} \end{align*}$ is an equivalence relation.

Proof:
Given that $(G,\circ)$ is a group and $\begin{align*} \rho=\set{(x,y)\in G\times G: y \text{ is a conjugate of } x} \end{align*}$ To prove $\rho$ is an equivalence relation

To prove $(a,a)\in \rho$
Let $e$ is the identity element of $G$ then $\begin{align*} & a=e\circ a \circ e^{-1}\\ \implies & a \text{ conjugate of } a \\ \implies & (a,a)\in \rho \end{align*}$
To prove if $(a,b)\in \rho$ then $(b,a)\in \rho$
Let $\begin{align*} & (a,b)\in \rho \\ \implies & b \text{ conjugate of } a \\ \implies & b=x\circ a \circ x^{-1} \text{ for some } x\in G\\ \implies & x^{-1} \circ b \circ x= a \implies & a \text{ conjugate of } b \\ \implies & (b,a)\in \rho \end{align*}$
To prove if $(a,b)\in \rho$ and $(b,c)\in \rho$ then $(a,c)\in \rho$
Let $\begin{align*} & (a,b)\in \rho \text{ and } (b,c)\in \rho\\ \implies & b=x\circ a \circ x^{-1} \text{ and } c=y \circ b \circ y^{-1} \text{ for some } x,y\in G\\ \implies & c=y \circ \big( x\circ a \circ x^{-1} \big) \circ y^{-1} \\ \implies & c=\big(y \circ x \big) \circ a \circ \big(x^{-1} \circ y^{-1} \big)\\ \implies & c=\big(y \circ x \big) \circ a \circ \big( y \circ x \big)^{-1}\\ \implies & c \text{ conjugate of } a \\ \implies & (a,c)\in \rho \end{align*}$

Hence $\rho$ is an equivalence relation.

Applications

Group Actions are crucial in a wide range of applications across mathematics and science. In geometry, group actions help classify shapes and structures based on their symmetries. In physics, they are used to study conservation laws and quantum mechanics. Group actions also play a role in coding theory, providing solutions to problems in communication systems. For further study, explore Relations and Ring Theory.

Conclusion

The study of Normalizer and Conjugate deepens the understanding of group structures and their symmetries. These concepts bridge the gap between theoretical and applied Mathematics, making them indispensable in the exploration of group properties. Their applications in symmetry, geometry, and algebra reaffirm their importance in Abstract Algebra.

References

Introduction to Group Theory by Benjamin Steinberg
Topics in Group Theory by Geoffrey Smith
Abstract Algebra by David S. Dummit and Richard M. Foote
Algebra by Michael Artin
Symmetry and Group Theory by Mark A. Armstrong

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FAQs

What is the normalizer of an element in a group?
The normalizer of an element is the set of all group elements that commute with it under conjugation.
What is the conjugate of an element?
The conjugate of an element is derived by applying a group operation that transforms the element within the group.
Why is the normalizer important in group theory?
It identifies the largest subgroup where a given subset remains stable under conjugation.
How are conjugates used in abstract algebra?
They are used to study equivalence and symmetry among elements of a group.
What are the applications of the normalizer?
It is applied in solving problems related to subgroup structures and invariant subsets.
How is the conjugate calculated?
The conjugate of an element $a$ by $g$ is calculated as $gag^{-1}$ .
What is the relationship between normalizers and centralizers?
The centralizer is a subset of the normalizer where all elements commute.
Can normalizers and conjugates simplify group theory problems?
Yes, they simplify complex problems by organizing elements based on symmetry and invariance.
Where can I find exercises on normalizers and conjugates?
Practice problems are available in Mathematics Questions.
How are normalizers and conjugates taught in universities?
These concepts are part of courses in Group Theory and abstract algebra.