p-group and p-subgroup - Definitions and Key theorems
p-group and p-subgroup
p-group and p-subgroup are essential concepts in Abstract Algebra. Historically, these concepts have helped in understanding the structure and behavior of groups under prime power orders. The significance of p-group and p-subgroup lies in their application in advanced mathematical theories and problem-solving.
What You Will Learn?
In this post, you will explore:
- Definition: p-group
- Definition: p-subgroup
- Theorem-1: Let \((G,\circ)\) be a non trivial group. Then \(G\) is a finite p-group if and only if \(\big|G\big|=p^{k}\) for some integer \(k\).
- Theorem-2: Let \((G,\circ)\) be a non trivial group. Then \(G\) is a finite p-group if and only if \(\big|Z(G)\big|\gt 1\).
- Theorem-3: Let \((G,\circ)\) be a group of order \(p^{2}\), where \(p\) is a prime. Then \(G\) is commutative.
Things to Remember
- Set Theory
- Relations
- Mappings
- Group Theory
Introduction
p-group and p-subgroup play a crucial role in Mathematics. A p-group is a group where the order of every element is a power of a prime number, while a p-subgroup is a subgroup that satisfies similar conditions within a larger group. These concepts are foundational in Abstract Algebra and provide valuable insights into group properties.
p-group
Definition:
Let \((G,\circ)\) be a group and \(p\) be a prime. \(G\) is said to be a p-group if the order of each element of \(G\) is a power of \(p\).
p-subgroup
Definition:
Let \((G,\circ)\) be a group and \(H\) be a subgroup of \(G\). Let \(p\) be a prime. \(H\) is said to be a p-subgroup if \(H\) is a p-group.
Theorem-1
Statement:
Let \((G,\circ)\) be a non trivial group. Then \(G\) is a finite p-group if and only if \(\big|G\big|=p^{k}\) for some integer \(k\).
Proof:
Given that \((G,\circ)\) is a non trivial group.
- Let \(G\) be a fnite p-group.
To prove \(\big|G \big|=p^{k}\) for some integer \(k\)
If possible let \(q\) be a prime such that \(q\big|~\big|G \big| \) and \(q\ne p \).
Then by Cauchy’s Theorem
\(\exists\) an element \(a\in G\), such that \(\big|a \big|=q \).
A contradiction, since \(G\) is a p-group.
Therefore \(p\) is the only prime divisor of \(\big|G \big|\).
Hence \(\big|G \big|=p^{k}\) for some integer \(k\). - Let \(\big|G \big|=p^{k}\) for some integer \(k\).
To prove \(G\) is a finite p-group
Then by Lagrange’s Theorem
each element \(a\in G\) has a power of \(p\).
Therefore \(G\) is a finite p-group.
Theorem-2
Statement:
Let \((G,\circ)\) be a non trivial group. Then \(G\) is a finite p-group if and only if \(\big|Z(G)\big|\gt 1\).
Proof:
Given that \((G,\circ)\) is a non trivial group. Then \(\big|G\big|\gt 1\).
Let \(G\) be a fnite p-group.
To prove \(\big|Z(G)\big|\gt 1\)
- Case-1: Let \(Z(G)=G\)
Since \(\big|G\big|\gt 1\) then \(\big|Z(G)\big|\gt 1\). - Case-2: Let \(Z(G)\subset G\)
Then \(G-Z(G)\ne \phi \). Let \begin{align*} & a\in G-Z(G) \\ \implies & a\notin Z(G)\\ \implies & C(a)\subset G \\ \implies & C(a)\text{ is a p-subgroup }~\big[\because G \text{ is a p-group }\big]\\ \implies & p\big|~\big[G:C(a)\big]\\ \implies & p\big|~\displaystyle\sum_{a\notin Z(G)}^{}\big[G:C(a)\big] \end{align*} Then from the Class Equation,
\(\big|G\big|=\big|Z(G)\big|+\displaystyle\sum_{a\notin Z(G)}^{}\big[G:C(a)\big]\)
we have, \begin{align*} p\big|~\big|Z(G)\big|~\big[\because p\big|~\big|G\big| \text{ and } p\big|~\displaystyle\sum_{a\notin Z(G)}^{}\big[G:C(a)\big] \big] \end{align*} Hence \(\big|Z(G)\big|\gt 1\).
Theorem-3
Statement:
Let \((G,\circ)\) be a group of order \(p^{2}\), where \(p\) is a prime. Then \(G\) is commutative.
Proof:
Given that \((G,\circ)\) is a group of order \(p^{2}\) where \(p\) is a prime.
To prove \(G\) is commutative.
Since \(p\) is a prime then \(\big|G\big|\gt 1 \) implies \(\big|Z(G)\big|\gt 1 \).
Then by Lagrange’s Theorem,
\(\big|Z(G)\big|=p\) or \(\big|Z(G)\big|=p^{2}\)
If possible let,
\begin{align*}
& \big|Z(G)\big|=p \\
\implies & 1 \gt \big|Z(G)\big| \gt \big|G\big| \\
\implies & Z(G)\subset G \\
\implies & G-Z(G)\ne \phi \\
\end{align*}
Let
\begin{align*}
& a\in G-Z(G) \\
\implies & a\notin Z(G)\\
\implies & Z(G) \subset C(a) \\
\implies & \big|C(a)\big|=p^{2}\\
\implies & C(a)=G \\
\implies & a\in Z(G) \\
\end{align*}
A contradiction. since \(a\notin Z(G) \).
Our assumption is wrong.
Therefore
\begin{align*}
& \big|Z(G)\big|=p^{2}\\
\implies & Z(G)=G
\end{align*}
Hence \(G\) is commutative.
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
p-group and p-subgroup remain indispensable in the study of Abstract Algebra. Their ability to simplify complex group structures highlights their importance in mathematical research and education, particularly in Mathematics.
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 p-group?
A p-group is a group where the order of every element is a power of a prime number. - What is a p-subgroup?
A p-subgroup is a subgroup within a group where all elements have orders that are powers of a specific prime number. - Why are p-groups important?
They provide insights into the structure of groups, particularly in finite group theory. - How are p-groups applied in Abstract Algebra?
They are used in the analysis of finite groups and in proving key theorems such as Sylow theorems. - Can a group have more than one p-subgroup?
Yes, a group can have multiple p-subgroups, and their properties are studied in detail in group theory. - What is the significance of p-groups in Mathematics?
They simplify the classification of groups and aid in understanding complex group structures. - Are p-groups always finite?
No, p-groups can be infinite, but they are primarily studied in the context of finite groups. - Where can I find examples of p-groups?
Examples can be found in Mathematics Questions and Abstract Algebra Questions. - What is the relation between p-groups and Sylow Theorems?
Sylow theorems provide conditions for the existence and uniqueness of p-subgroups in finite groups. - How do p-subgroups aid in solving mathematical problems?
They simplify the analysis of group structures, making complex problems more approachable.
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