Conjugacy Class and Class Equation
Conjugacy Class and Class Equation
Conjugacy Class and Class Equation are vital concepts in Mathematics and Abstract Algebra. These topics have a rich history in the study of groups, first introduced to analyze symmetry in mathematical and physical systems. Their applications in Group Theory are widely recognized for exploring group structure and properties.
What You Will Learn?
In this post, you will explore:
- Definition: Conjugacy Class
- Theorem-1: Let \((G,\circ)\) be a group and \(\rho=\set{(x,y)\in G \times G: y \text{ is a conjugate of } x }\) be the equivalence relations and \( a\in G\). Then \begin{align*} \big|[a]\big|=\big[G:C(a) \big]\end{align*}.
In addition if \(G\) is finite then \begin{align*} \big|G\big|=\displaystyle\sum_{a}^{}\big[G:C(a) \big]\end{align*} where the summation is over a complete set of distinct conjugacy classes. - Class Equation: Let \((G,\circ)\) be a finite group \begin{align*} \big|G\big|=\big|Z(G)\big|+\displaystyle\sum_{a\bcancel{\in} Z(G)}^{}\big[G:C(a) \big]\end{align*} where the summation is over a complete set of distinct conjugacy classes.
Things to Remember
- Set Theory
- Relations
- Mappings
- Group Theory
Introduction
Conjugacy Class and Class Equation are fundamental tools in understanding group behavior in Abstract Algebra. A conjugacy class groups elements of a group that are similar under conjugation, while the class equation provides a breakdown of the group’s order into simpler components. These topics are integral to solving Abstract Algebra Questions and exploring advanced algebraic properties.
Conjugacy Class
Definition:
Let \((G,\circ)\) be a group and \(\rho=\set{(x,y)\in G \times G: y \text{ is a conjugate of } x }\) be the equivalence relations and \( a\in G\). Then the equivalence class of \([a]\) is called the conjugacy class of \(a\) in \(G\).
Theorem-1
Statement:
Let \((G,\circ)\) be a group and \(\rho=\set{(x,y)\in G \times G: y \text{ is a conjugate of } x }\) be the equivalence relations and \( a\in G\). Then \begin{align*} \big|[a]\big|=\big[G:C(a) \big]\end{align*}.
In addition if \(G\) is finite then \begin{align*} \big|G\big|=\displaystyle\sum_{a}^{}\big[G:C(a) \big]\end{align*}
where the summation is over a complete set of distinct conjugacy classes.
Proof:
Given that \((G,\circ)\) is a group \(\rho=\set{(x,y)\in G \times G: y \text{ is a conjugate of } x }\) is the equivalence relations and \( a\in G\).
To prove \( \big|[a]\big|=\big[G:C(a) \big] \)
Let \(T\) be the set of all distinct left cosets of \(C(a)\) in \(G\).
Then we have \(\big|T\big|=\big[G:C(a) \big]\).
Let us construct a mapping \(f:T\to [a]\) such that for \(x\in G\).
\begin{align*}
f(xC(a))=x\circ a \circ x^{-1}
\end{align*}
- To prove \(f\) is well defined.
Let \(xC(a), yC(a) \in T\) for some \(x,y\in G\) such that \begin{align*} & xC(a)=yC(a) \\ \implies & y^{-1}\circ x \in C(a) \\ \implies & \big( y^{-1}\circ x \big) \circ a= a \circ \big( y^{-1}\circ x \big)\\ \implies & y^{-1}\circ \big( x \circ a \big)= \big( a \circ y^{-1} \big)\circ x \\ \implies & x\circ a \circ x^{-1}= y\circ a \circ y^{-1}\\ \implies & f(xC(a))= f(yC(a)) \end{align*} \(\therefore \) \(f\) is well defined.
- To prove \(f\) is injective.
Let \(xC(a), yC(a) \in T\) for some \(x,y\in G\) such that \begin{align*} & f(xC(a))= f(yC(a)) \\ \implies & x\circ a \circ x^{-1}= y\circ a \circ y^{-1}\\ \implies & \big( y^{-1}\circ x \big) \circ a= a \circ \big( y^{-1}\circ x \big)\\ \implies & y^{-1}\circ x \in C(a) \\ \implies & xC(a)=yC(a) \end{align*} \(\therefore \) \(f\) is injective.
- To prove \(f\) is surjective.
Let \begin{align*} & z \in [a] \\ \implies & z \text{ is a conjugate of } a\\ \implies & \exists \text{ an element } w\in G \text{ such that } z=w\circ a \circ w^{-1}\\ \implies & z=f(wC(a))\\ \end{align*} \(\therefore wC(a)\) is a pre-image of \(z\) in \(T\). Since \(z\) is an arbitrary element of \([a]\), then each element of \([a]\) has a pre-image in \(T\).
\(\therefore \) \(f\) is surjective.
Therfore \(\big|T\big|=[a]\).
Hence \( \big|[a]\big|=\big[G:C(a) \big] \).
2nd Part-
Let \(G\) is finite. We have \(\rho\) is the equivalence relation on \(G\).
Therefore \(G=\cup_{a} [a]\) where the union runs over a complete set of distinct conjugacy classes. Then
\begin{align*}
& \big|G\big|=\displaystyle\sum_{a}^{} \big| [a]\big|\\
\implies \big|G\big|= \displaystyle\sum_{a}^{}\big[G:C(a) \big]\\
& \big[\because \big|[a]\big|=\big[G:C(a) \big] \big]
\end{align*}
where the summation is over a complete set of distinct conjugacy classes.
Hence the theorem is proved.
Class Equation
Statement:
Let \((G,\circ)\) be a finite group
\begin{align*}
\big|G\big|=\big|Z(G)\big|+\displaystyle\sum_{a\bcancel{\in} Z(G)}^{}\big[G:C(a) \big]\end{align*}
where the summation is over a complete set of distinct conjugacy classes.
Proof:
Given that \((G,\circ)\) is a finite group.
To prove \(\big|G\big|=\big|Z(G)\big|+\displaystyle\sum_{a\bcancel{\in} Z(G)}^{}\big[G:C(a) \big]\)
We have
\begin{align}
& \big|G\big|= \displaystyle\sum_{a}^{}\big[G:C(a) \big] \nonumber \\
\implies & \big|G\big|= \displaystyle\sum_{a\in Z(G)}^{}\big[G:C(a) \big]+\displaystyle\sum_{a\bcancel{\in} Z(G)}^{}\big[G:C(a) \big]
\end{align}
Now let
\begin{align}
& a\in Z(G) \nonumber \\
\implies & C(a)=G \nonumber \\
\implies & \big[G:C(a) \big]=1
\end{align}
Substituting (2) in (1), we get
\begin{align*}
& \big|G\big|= \displaystyle\sum_{a\in Z(G)}^{}1+\displaystyle\sum_{a\bcancel{\in} Z(G)}^{}\big[G:C(a) \big] \\
\implies & \big|G\big|= \big|Z(G)\big|+\displaystyle\sum_{a\bcancel{\in} Z(G)}^{}\big[G:C(a) \big]
\end{align*}
Hence the theorem is proved.
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 Conjugacy Class and Class Equation provides profound insights into the structural organization of groups. These concepts simplify the analysis of Mathematics and enable the classification of group elements effectively. Their importance in Abstract Algebra ensures their continued relevance in mathematical research and applications.
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 a conjugacy class?
A conjugacy class groups elements of a group that are equivalent under conjugation. - What is the class equation?
The class equation expresses the order of a group as the sum of the sizes of its conjugacy classes. - Why are conjugacy classes important?
They simplify group classification and analysis by grouping similar elements. - How is the class equation used?
It helps determine properties like group structure and possible subgroups. - What are the applications of conjugacy classes?
They are applied in symmetry analysis and group characterizations. - What is an example of a class equation?
For the symmetric group \( S_3 \), the class equation is \( |S_3| = 1 + 2 + 3 \). - How are conjugacy classes related to normal subgroups?
A normal subgroup is a union of conjugacy classes. - What is the center of a group in terms of conjugacy classes?
The center is the set of elements forming their own conjugacy class. - Where can I practice problems on these topics?
Practice problems are available on Mathematics Questions. - What courses include conjugacy classes and the class equation?
These are core topics in Group Theory and abstract algebra courses.
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