Sylow's Third Theorem
Sylow's Third Theorem
Sylow's Third Theorem holds significant importance in Mathematics, particularly in Abstract Algebra and Group Theory. Developed by Ludwig Sylow, the theorem provides a structured understanding of subgroup distribution within finite groups. This theorem, alongside the other Sylow theorems, plays a critical role in analyzing the properties and structures of groups.
What You Will Learn?
In this post, you will explore:
- Sylow’s Third Theorem: Let \((G,\circ)\) be a finite group of order \(p^{r}m\), where \(p\) be a prime and \(r,m\) be two positive integers and \(p,m\) be relatively prime. Then the number \(n_{p} \) of Sylow p-subgroups of \(G\) is \(1+kp\) for some non-negative integer \(k\) and \(n_{p}\big|p^{r}m \).
Things to Remember
- Set Theory
- Relations
- Mappings
- Group Theory
Introduction
Sylow’s Third Theorem begins by exploring the uniqueness of Sylow p-subgroups in group structures. This theorem finds applications in Mathematics and offers a foundation for tackling complex problems in Abstract Algebra. By leveraging the power of this theorem, mathematicians can classify finite groups effectively and understand their composition.
Sylow's Third Theorem
Statement:
Let \((G,\circ)\) be a finite group of order \(p^{r}m\), where \(p\) be a prime and \(r,m\) be two positive integers and \(p,m\) be relatively prime. Then the number \(n_{p} \) of Sylow p-subgroups of \(G\) is \(1+kp\) for some non-negative integer \(k\) and \(n_{p}\big|p^{r}m \).
Proof:
Given that \((G,\circ)\) is a finite group of order \(p^{r}m\), where \(p\) is a prime and \(r,m\) are positive integers and \(p,m\) are relatively prime.
Let \(n_{p}= \) the number of Sylow p-subgroups of \(G\).
- To prove \(n_{p}=1+kp \) for some non-negative integer \(k\)
Let \(S\) be the set of all Sylow p-subgroups of \(G\).
Then \(\big|S\big|=n_{p}\).
Let \(P\in S\).
First we prove \(P\) acts on S.- Let us construct a mapping \(\cdot:P\times S\to S \) such that for all \(a\in P, Q\in S,\) \begin{align*}a \cdot Q= aQa^{-1}
\end{align*}
- To prove \(\cdot \) is well defined.
Let \(a,b\in P\) and \(Q,R\in S\) such that \begin{align*} & \big(a,Q \big)=\big(b,R \big)\\ \implies & a=b,~ Q=R\\ \implies & aQa^{-1}=bRb^{-1}\\ \implies & a \cdot Q=b \cdot R \end{align*} Therefore \(\cdot \) is well defined. - Let \(a,b\in P\) and \(Q\in S\). Now
\begin{align*}
\big(a\circ b\big) \cdot Q & = \big(a\circ b\big) Q \big(a\circ b\big)^{-1}\\
& = \big(a\circ b\big) Q \big(b^{-1}\circ a\big)\\
& = a\big(b Q b^{-1}\big)a \\
& = a\cdot \big(b Q b^{-1}\big)\\
& = a\cdot \big(b \cdot Q \big)
\end{align*}
- Let \( e\) be the identity element of \(G\) and \(Q\in S\). Now
\begin{align*}
e \cdot Q= e Q e^{-1} = Q
\end{align*}
- To prove \(\cdot \) is well defined.
Let \begin{align*} & S_{0}=\set{Q\in S: a \cdot Q= Q ~\forall~ a\in P} \\ \implies & S_{0}=\set{Q\in S: a Q a^{-1}= Q ~\forall~ a\in P} \end{align*} Since \begin{align*} & P\in S \text{ and } a P a^{-1}= P ~\forall~ a\in P\\ \implies & P\in S_{0}\\ \implies & S_{0}\ne \phi\\ \end{align*} Let \begin{align*} & R\in S_{0} \\ \implies & a R a^{-1}= R ~\forall~ a\in P\\ \implies & P \subseteq N(R) \end{align*} Therefore \(P\) and \(R\) are Sylow p-subgroups of \(N(R)\).
Then Sylow’s Second theorem, \(\exists\) a \(\alpha \in N(R)\) such that \begin{align*} & a R a^{-1}= P \\ \implies & R= P ~\because~a R a^{-1}= R~\forall~a\in P\\ \end{align*} Since \(R\in S_{0}\) is arbitrary then \begin{align*} & S_{0}=\set{P}\\ \implies & \big|S_{0} \big|=1 \end{align*} Since \(P\) is a Sylow p-subgroup of \(G\) then \(\big|P\big|=p^{r} \). Therefore \begin{align*} & \big|S\big|=\big|S_{0}\big|\big(~mod~p\big) \\ \implies & \big|S\big|=1\big|\big(~mod~p\big) \\ \implies & \big|S\big|=1+kp ~\text{ where } k \text{ is a non-negative integer }\\ \implies & n_{p}=1+kp \\ \end{align*}
- Let us construct a mapping \(\cdot:P\times S\to S \) such that for all \(a\in P, Q\in S,\) \begin{align*}a \cdot Q= aQa^{-1}
\end{align*}
- To prove \(n_{p}\big|p^{r}m \)
Let us construct a mapping \(\cdot:G\times S\to S \) such that for all \(a\in G, Q\in S,\) \begin{align*}a \cdot Q= aQa^{-1} \end{align*} Then by previous arguments, \(G\) acts on \(S\).
Since \(\big|G\big|=p^{r}m\) then \(G\) has Sylow p-subgroup.
Then \(S\ne\phi\).
Let \(P\in S\).
-
Let \(R\in S\).
\(\implies R\) is a Sylow p-subgroup.
\(\implies R\) and \(P\) are conjugates, using Sylow’s Second theorem.
\(\implies \exists\) an element \(\beta\in G\) such that \(\beta P \beta^{-1}=R\)
\(\implies R\in \big[P\big]\)
\(\implies S\subseteq \big[P\big]\)
\(\implies \big[P\big]=S~\because~\big[P\big]\subseteq S\)
Let \(A\) is a subset of \(S\) containing exactly one elements from each orbit. Then \(\big|A\big|=1\).
We have \begin{align*} \big|S\big|=\displaystyle\sum_{\alpha\in A}^{}\big[G:G_{\alpha} \big] \end{align*} where \(A\) is a subset of \(S\) containing exactly one elements from each orbit.
\begin{align*} & \big|S\big|=\big[G:G_{\alpha} \big]~\because~\big|A\big|=1 \\ \implies & \big|S\big|~\Big|~\big|G\big|~\because~\big[G:G_{\alpha} \big]~\Big|~\big|G\big| \\ \implies & n_{p}~\big|~p^{r}m \end{align*} -
Let \(R\in S\).
Applications
Sylow’s Third Theorem has numerous applications in solving advanced problems in Abstract Algebra and Group Theory. It provides insights into the existence and quantity of p-subgroups, simplifying the analysis of symmetry groups and helping identify normal subgroups. Applications of this theorem extend to number theory and physics, where symmetry principles are integral.
Conclusion
Sylow’s Third Theorem is essential for understanding group structure and subgroup characteristics in Mathematics. This theorem continues to guide researchers in tackling problems within Abstract Algebra and contributes to advancements in mathematical sciences.
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 Sylow’s Third Theorem?
It describes the conditions under which Sylow p-subgroups exist and the number of such subgroups in a finite group. See Abstract Algebra. - Why is Sylow’s Third Theorem important?
It provides a structured way to analyze finite groups and their p-subgroups. See Group Theory. - How is Sylow’s Third Theorem applied?
It is used to classify finite groups and explore normal subgroup existence. Visit Mathematics problems for examples. - Who developed Sylow’s Theorems?
They were developed by Ludwig Sylow, a Norwegian mathematician. - What are Sylow p-subgroups?
They are subgroups of a finite group with orders that are powers of a prime number p. See Group Theory. - What does Sylow’s Third Theorem state?
It specifies conditions for the number and distribution of Sylow p-subgroups. - Is Sylow’s Third Theorem used in physics?
Yes, it is applied in symmetry analysis in physical systems. - Are Sylow’s Theorems foundational?
Yes, they are foundational in Abstract Algebra. - How are Sylow’s Theorems related to Lagrange’s Theorem?
They extend the principles of subgroup orders in finite groups. See Mathematics. - Where can I study problems related to Sylow’s Third Theorem?
Visit Abstract Algebra problems for practice questions.
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