Group Actions Notes - Definitions and Key Theorems
Group Actions
Group Actions provide an in-depth overview of how groups interact with mathematical structures. Historically, the concept of group actions emerged as a way to study symmetries and their applications in geometry, physics, and algebra. The importance of group actions lies in their ability to bridge the abstract theory of groups with practical applications, making this topic essential for students studying Abstract Algebra.
What You Will Learn?
In this post, you will explore:
- Definition: Group Actions
- Definition: G-Set
- Theorem-1: Let \((G,\circ )\) be a group and \(S\) be a non-empty set and \(S\) be a G-Set. Then prove that the relation \(\rho\) on \(S\) defined by \(a\rho b\iff g\cdot a=b \) for some \(g\in G\) is an equivalence relation.
- Definition: Orbits
- Definition: Stabilizer
- Theorem-2: Let \((G,\circ )\) be a group and \(S\) be a non-empty set and \(S\) be a G-Set. If \(a\in S\) then the stabilizer of \(a\) is a subgroup of \(G\).
Things to Remember
- Set Theory
- Relations
- Mappings
- Group Theory
Introduction
Group Actions introduce a critical concept in Group Theory that explains how a group operates on a set or a mathematical structure. This topic is fundamental to understanding the relationships between algebraic structures and their symmetries, providing a powerful tool for solving problems in Mathematics. Group actions simplify the study of transformations, making them indispensable in Group Theory and beyond.
Group Action
Definition:
Let \((G,\circ )\) be a group and \(S\) be a non-empty set. The (left) action of \(G\) on \(S\) is a mapping \(\cdot :G\times S \to S\) such that
- \(\left(g_{1}\circ g_{2} \right)\cdot s=g_{1}\cdot \left(g_{2}\cdot s \right)~\forall~g_{1},g_{2}\in G,~\forall~s\in S \)
- \(e\cdot s=s\) where \(e\) is the identity element of \(G\)
G-Set
Definition:
Let \((G,\circ )\) be a group and \(S\) be a non-empty set. Then \(S\) is said to be a G-Set if there exists a action of \(G\) on \(S\).
Theorem-1
Statement:
Let \((G,\circ )\) be a group and \(S\) be a non-empty set and \(S\) be a G-Set. Then prove that the relation \(\rho\) on \(S\) defined by \(\forall~a,b \in S \), \(a\rho b\iff g\cdot a=b \) for some \(g\in G\) is an equivalence relation.
Proof:
Given that \((G,\circ )\) is a group and \(S\) is a non-empty set and \(S\) is a G-Set.
Let \(\cdot :G\times S \to S\) be an action of \(G\) on \(S\).
Let \(\rho\) be a relation on \(S\) defined by \(\forall~a,b \in S \), \(a\rho b\iff g\cdot a=b \) for some \(g\in G\).
To prove that \(\rho\) is an equivalence relation.
- To prove \(\rho\) is reflexive.
Let \(a\in S\).
And \(e\in G\) be the identity element. Then we have \(e\cdot a=a\).
\(\implies a \rho a \)
\(\therefore\) \(\rho\) is reflexive.
- To prove \(\rho\) is symmetric.
Let \(a,b\in S\) such that \( a \rho b \).
Then there exists an element \(g\in G\) such that \begin{align*} & b=g\cdot a\\ \implies & g^{-1}\cdot b=g^{-1}\cdot\big(g\cdot a\big)\\ \implies & g^{-1}\cdot b=\big(g^{-1}\circ g\big)\cdot a\\ \implies & g^{-1}\cdot b=e\cdot a\\ \implies & g^{-1}\cdot b=a\\ \implies & b\rho a ~\big[\because g^{-1}\in G] \end{align*} \(\therefore\) \(\rho\) is symmetric.
- To prove \(\rho\) is transitive.
Let \(a,b,c\in S\) such that \( a \rho b \) and \( b \rho c \).
Then there exists an element \(g_{1},g_{2}\in G\) such that \(b=g_{1}\cdot a\) and \(c=g_{2}\cdot b\). Now \begin{align*} & c=g_{2}\cdot b\\ \implies & c=g_{2}\cdot \big( g_{1}\cdot a \big)\\ \implies & c=\big( g_{2}\circ g_{1}\big) \cdot a \\ \implies & a\rho c ~\big[\because g_{2}\circ g_{1}\in G\big] \end{align*} \(\therefore\) \(\rho\) is transitive.
Hence \(\rho\) is an equivalence relation.
Orbits
Definition:
Let \((G,\circ )\) be a group and \(S\) be a non-empty set and \(S\) be a G-Set. And \(\rho\) be the equivalence relation defined by \(\forall~a,b \in S \), \(a\rho b\iff g\cdot a=b \) for some \(g\in G\). Then the equivalence classes on \(S\) are called the orbits of \(G\) on \(S\). And is denoted by for \(a\in S\), \([a]=\set{x\in S:~ g\cdot a=x,~ g\in G}\).
Stabilizer
Definition:
Let \((G,\circ )\) be a group and \(S\) be a non-empty set and \(S\) be a G-Set. Then for \(a\in S\), the subset \(G_{a}=\set{g\in G:~ g\cdot a=a}\) is called the stabilizer of \(a\).
Theorem-2
Statement:
Let \((G,\circ )\) be a group and \(S\) be a non-empty set and \(S\) be a G-Set. If \(a\in S\) then the stabilizer of \(a\) is a subgroup of \(G\).
Proof:
Given that \((G,\circ )\) is a group and \(S\) is a non-empty set and \(S\) is a G-Set.
Let \(\cdot :G\times S \to S\) be an action of \(G\) on \(S\).
Let \(a\in S\).
To prove \(G_{a}=\set{g\in G:~ g\cdot a=a}\) is a subgroup of \(G\).
- Let \(e\in G\) be the identity element.
Then \(e\cdot a= a\).
\(\therefore G_{a}\ne \phi\).
- Let \(g_{1},g_{2}\in G_{a}\)
Then we have \(g_{1}\cdot a=a \) and \(g_{2}\cdot a=a \). Now \begin{align*} &\big( g_{1}\circ g_{2}\big)\cdot a= g_{1}\cdot \big(g_{2}\cdot a\big)\\ \implies &\big( g_{1}\circ g_{2}\big)\cdot a= g_{1}\cdot a ~\big[\because ~g_{2}\cdot a=a\big]\\ \implies &\big( g_{1}\circ g_{2}\big)\cdot a= a ~\big[\because ~g_{1}\cdot a=a\big]\\ \implies & g_{1}\circ g_{2}\in G_{a} \end{align*} \(\therefore g_{1}\circ g_{2}\in G_{a}~\forall g_{1},g_{2}\in G_{a}\).
- Let \(g\in G_{a}\). Now
\begin{align*}
& g\cdot a= a\\
\implies & g^{-1}\cdot\big(g\cdot a\big)=g^{-1}\cdot a\\
\implies & \big(g^{-1}\cdot g\big)\cdot a=g^{-1}\cdot a\\
\implies & e\cdot a=g^{-1}\cdot a\\
\implies & a=g^{-1}\cdot a\\
\implies & g^{-1}\in G_{a}
\end{align*}
\(\therefore g^{-1}\in G_{a}~\forall g\in G_{a}\).
Hence \(G_{a}\) is a subgroup of \(G\).
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
Group Actions are a vital resource for students and researchers exploring the interaction between groups and mathematical structures. By studying group actions, one gains insights into symmetries, transformations, and their profound applications 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 a group action?
A group action describes how a group operates on a set, preserving the group structure through transformations. -
Why are group actions important?
Group actions are important for studying symmetries, transformations, and their applications in various fields. -
What is an orbit in group actions?
An orbit is the set of elements that can be reached by applying all the group elements to a single point in the set. -
What is a stabilizer in group actions?
A stabilizer is the subset of group elements that leave a point in the set unchanged under the group action. -
How are group actions applied in geometry?
Group actions are used to classify shapes and structures based on their symmetries and transformations. -
What is Burnside’s Lemma?
Burnside’s Lemma is a mathematical tool used to count distinct objects under group actions. -
What are transitive actions?
Transitive actions occur when a group action has a single orbit, connecting all elements of the set. -
What are free actions?
Free actions occur when no element other than the identity fixes any point in the set under the group action. -
How do group actions relate to coding theory?
Group actions provide solutions to problems in communication systems by analyzing symmetries. -
What are some advanced applications of group actions?
Group actions are used in studying Galois theory, quantum mechanics, and mathematical physics.
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