Characteristic Subgroup in Group Theory: Definition and Key Theorems
Characteristic Subgroup in Group Theory
This article explores the concept of characteristic subgroup in group theory, focusing on their definition, properties, and key theorems. A characteristic subgroup is defined as a subgroup that remains invariant under all automorphisms of its parent group. This characteristic property makes them essential in the study of symmetry and structure in mathematical groups, a topic that plays a significant role in various fields, including Abstract Algebra and higher mathematics education. The article discusses the historical context of characteristic subgroups, highlights contributions from notable mathematicians, and addresses their importance in competitive exams and university courses. Applications of characteristic subgroups are also examined in areas like physics, cryptography, and economics, providing readers with a comprehensive understanding of their relevance in both theoretical and practical contexts.
What You Will Learn?
- Definition: Characteristic Subgroup
- Theorems:
- Prove that every characteristic subgroup of group \(\pmb{(G,\cdot)}\) is a normal subgroup of \(\pmb{G}\). But the converse is not true.
- Let \(\pmb{(G,\cdot)}\) be a group. Then prove that \(\pmb{Z(G)}\) is a characteristic subgroup of a group \(\pmb{G}\).
- Let \(\pmb{(G,\cdot)}\) be a group and \(\pmb{H}\) and \(\pmb{K}\) be two characteristic subgroups of \(\pmb{G}\). Then show that \(\pmb{HK}\) and \(\pmb{H\cap K}\) are two characteristic subgroups of \(\pmb{G}\).
- Let \(\pmb{(G,\cdot)}\) be a group and \(\pmb{H}\) and \(\pmb{K}\) be two subgroups of \(\pmb{G}\) such that \(\pmb{H\subseteq K}\). If \(\pmb{K}\) is a characteristic subgroup of \(\pmb{G}\) and \(\pmb{H}\) is a characteristic subgroup of \(\pmb{K}\) then \(\pmb{H}\) is a characteristic subgroup of \(\pmb{G}\).
- If \(\pmb{(G,\cdot)}\) be a cyclic group then every subgroup of \(\pmb{G}\) is a characteristic subgroup of \(\pmb{G}\).
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
In the realm of group theory, a characteristic subgroup holds a unique place in defining properties that are intrinsic to the structure of mathematical groups. Introduced as a way to study deeper symmetry, characteristic subgroups have become fundamental in Group Theory and Abstract Algebra, with wide-reaching applications across B.Sc. Mathematics and M.Sc. Mathematics under the NEP (National Education Policy). The study of characteristic subgroups is integral for students preparing for competitive exams and university-level assessments.
Historically, the concept of a characteristic subgroup was developed to understand the inherent structures of groups and has been studied extensively by prominent mathematicians, including Évariste Galois and Felix Klein. Understanding characteristic subgroups is crucial as they help reveal properties preserved under automorphisms, which is essential in advanced group analysis and problem-solving in group theory.
Characteristic Subgroup
Definition:
Let \(\pmb{(G,\cdot)}\) be a group and \(\pmb{H}\) be a subgroup of \(\pmb{G}\). \(\pmb{H}\) is said to be a characteristic subgroup of \(\pmb{G}\) if \(\pmb{f(H)\subseteq H~\forall~f\in Aut(G)}\).
Theorem-1
Statement:
Prove that every characteristic subgroup of group \(\pmb{(G,\cdot)}\) is a normal subgroup of \(\pmb{G}\). But the converse is not true.
Proof:
Let \(\pmb{(G,\cdot)}\) be a group and \(\pmb{H}\) be a characteristic subgroup of \(\pmb{G}\).
To prove \(\pmb{H}\) is normal a subgroup of \(\pmb{G}\).
Since \(\pmb{H}\) is a characteristic subgroup of \(\pmb{G}\), then we have \(\pmb{f(H)\subseteq H~\forall~f\in Aut(G)}\).
Let \(\pmb{h\in H}\) and \(\pmb{g\in G}\)
\(\pmb{\implies \theta_{g}\in Aut(G) }\) where \(\pmb{\theta_{g}: H\to H }\) such that \(\pmb{\theta_{g}(x)=g\cdot x \cdot g^{-1}~\forall~x\in H}\)
\(\pmb{\implies g\cdot h \cdot g^{-1} \in H}\) since \(\pmb{h\in H}\)
Therefore \(\pmb{H}\) is normal a subgroup of \(\pmb{G}\).
Now to prove that the converse is not true.
Consider Klein’s 4-group \(\pmb{\left( K_{4} \right)}\).
Then we have \(\pmb{ K_{4}=\{e~,~a,~b~,~a\cdot b \} }\) with \(\pmb{a^{2}=e }\), \(\pmb{b^{2}=e }\) and \(\pmb{a \cdot b=b \cdot a }\). Also \(\pmb{o(a)=o(b)=o(a \cdot b)=2 }\).
Let consider a subgroup \(\pmb{H=\{e,a \}}\).
First we prove that \(\pmb{H}\) is normal in \(\pmb{K_{4}}\).
Since\(\pmb{K_{4}}\) is abelian group therefore \(\pmb{H}\) is normal in \(\pmb{K_{4}}\). Since every subgroup of an abelian group is normal.
Now consider a mapping \(\pmb{\phi:K_{4}\to K_{4}}\) such that \(\pmb{\phi= \begin{pmatrix}
e & a & b & a\cdot b \\
e & b & a & a\cdot b
\end{pmatrix} }\)
Clearly \(\pmb{\phi}\) is an isomorphism i.e., \(\pmb{\phi\in Aut(K_{4})}\).
But \(\pmb{\phi(H)=\{\phi(e)~,~\phi(a) \}=\{e~,~b \}\cancel{\subseteq}H }\)
Hence \(\pmb{H}\) is not a characteristic subgroup of \(\pmb{K_{4}}\).
Theorem-2
Statement:
Let \(\pmb{(G,\cdot)}\) be a group. Then prove that \(\pmb{Z(G)}\) is a characteristic subgroup of a group \(\pmb{G}\).
Proof:
Given that \(\pmb{(G,\cdot)}\) is a group.
To prove \(\pmb{Z(G)=H}\)(say) is a characteristic subgroup of a group \(\pmb{G}\).
Let \(\pmb{\phi\in Aut(G) }\).
Then we have to prove \(\pmb{\phi(H)\subseteq H }\).
Let \(\pmb{x\in \phi(H) }\)
Let \(\pmb{g\in G }\)
Then \(\pmb{ \exists~ y\in H~,~z \in G}\) such that \(\pmb{\phi(y)=x }\) and \(\pmb{\phi(z)=g }\). Since \(\pmb{\phi }\) is an isomorphism on \(\pmb{G}\).
Now \( \pmb{x \cdot g= \phi(y)\cdot \phi(z)}\)
\(\implies \pmb{x \cdot g= \phi(y \cdot z)}\) since \(\pmb{ \phi}\) is a homomorphism.
\(\implies \pmb{x \cdot g= \phi(z \cdot y)}\) since \(\pmb{ y\in H}\)
\(\implies \pmb{x \cdot g= \phi(z) \cdot \phi(y)}\) since \(\pmb{ \phi}\) is a homomorphism.
\(\implies \pmb{x \cdot g= g \cdot x}\)
Therefore \(\pmb{x \cdot g= g \cdot x ~\forall~g\in G}\)
\(\implies \pmb{x\in H }\)
Therefore \(\pmb{\phi(H)\subseteq H }\).
Hence \(\pmb{Z(G)}\) is a characteristic subgroup of a group \(\pmb{G}\).
Theorem-3
Statement:
Let \(\pmb{(G,\cdot)}\) be a group and \(\pmb{H}\) and \(\pmb{K}\) be two characteristic subgroups of \(\pmb{G}\). Then show that \(\pmb{HK}\) and \(\pmb{H\cap K}\) are two characteristic subgroups of \(\pmb{G}\).
Proof:
Given that \(\pmb{(G,\cdot)}\) is a group and \(\pmb{H}\) and \(\pmb{K}\) are two characteristic subgroups of \(\pmb{G}\).
To prove \(\pmb{HK}\) and \(\pmb{H\cap K}\) are two characteristic subgroups of \(\pmb{G}\).
Let \(\pmb{\phi\in Aut(G)}\).
- First we prove that \(\pmb{\phi(HK)\subseteq HK}\)
Let \(\pmb{x\in \phi(HK)}\)
Then \(\pmb{\exists~y\in HK}\) such that \(\pmb{\phi(y)=x}\)
\(\implies \pmb{y=h_{1}\cdot k_{1}}\) for some \(\pmb{h_{1}\in H~,~k_{1}\in K }\)
\(\implies \pmb{\phi(y)=\phi(h_{1}\cdot k_{1}) }\)
\(\implies \pmb{x=\phi(h_{1}) \cdot \phi(k_{1}) }\) since \(\pmb{\phi}\) is a homomorphism
\(\implies \pmb{x=h_{2}\cdot k_{2}}\) since \(\pmb{\phi(H)\subseteq H}\) and \(\pmb{\phi(K)\subseteq K}\) where \(\pmb{\phi(h_{1})=h_{2}}\) and \(\pmb{\phi(k_{1})=k_{2}}\).
\(\implies \pmb{x\in HK}\)
Therefore \(\pmb{\phi(HK)\subseteq HK}\).
Hence \(\pmb{HK}\) is a characteristic subgroup of \(\pmb{G}\). - Now we prove that \(\pmb{\phi(H\cap K)\subseteq H\cap K}\)
Let \(\pmb{z\in \phi(H\cap K)}\)
Then \(\pmb{\exists~w\in H\cap K}\) such that \(\pmb{\phi(w)=z}\)
\(\implies \pmb{w\in H}\) and \(\pmb{w\in K}\)
\(\implies \pmb{w=h_{3}}\) and \(\pmb{w=k_{3}}\) for some \(\pmb{h_{3}\in H~,~k_{3}\in K }\)
\(\implies \pmb{\phi(w)=\phi(h_{3})}\) and \(\pmb{\phi(w)=\phi(k_{3})}\)
\(\implies \pmb{ z=h_{4}}\) and \(\pmb{z=k_{4}}\) since \(\pmb{\phi(H)\subseteq H}\) and \(\pmb{\phi(K)\subseteq K}\) where \(\pmb{\phi(h_{3})=h_{4}}\) and \(\pmb{\phi(k_{3})=k_{4}}\).
\(\implies\pmb{ z\in H}\) and \(\pmb{z\in K}\)
\(\implies \pmb{z\in H\cap K}\)
Therefore \(\pmb{\phi(H\cap K)\subseteq H\cap K}\).
Hence \(\pmb{H\cap K}\) is a characteristic subgroup of \(\pmb{G}\).
Theorem-4
Statement:
Let \(\pmb{(G,\cdot)}\) be a group and \(\pmb{H}\) and \(\pmb{K}\) be two subgroups of \(\pmb{G}\) such that \(\pmb{H\subseteq K}\). If \(\pmb{K}\) is a characteristic subgroup of \(\pmb{G}\) and \(\pmb{H}\) is a characteristic subgroup of \(\pmb{K}\) then \(\pmb{H}\) is a characteristic subgroup of \(\pmb{G}\).
Proof:
Given that \(\pmb{(G,\cdot)}\) is a group and \(\pmb{H}\) and \(\pmb{K}\) are two subgroups of \(\pmb{G}\) such that \(\pmb{H\subseteq K}\).
Let \(\pmb{K}\) be a characteristic subgroup of \(\pmb{G}\) and \(\pmb{H}\) be a characteristic subgroup of \(\pmb{K}\).
To prove \(\pmb{H}\) is a characteristic subgroup of \(\pmb{G}\).
Let \(\pmb{\phi\in Aut(G)}\)
We have \(\pmb{\phi(K)\subseteq K}\) since \(\pmb{K}\) be a characteristic subgroup of \(\pmb{G}\).
Since \(\pmb{\phi}\) is an automorphism then the restriction of \(\pmb{\phi}\) to \(\pmb{K}\) i.e., \(\pmb{\phi|_{K}:K\to K}\) is also an automorphism.
Then \(\pmb{\phi|_{K}(H)\subseteq H}\) since \(\pmb{H}\) be a characteristic subgroup of \(\pmb{K}\).
Therefore \(\pmb{\phi(H)\subseteq H}\).
Hence \(\pmb{H}\) is a characteristic subgroup of \(\pmb{G}\).
Theorem-5
Statement:
If \(\pmb{(G,\cdot)}\) be a cyclic group then every subgroup of \(\pmb{G}\) is a characteristic subgroup of \(\pmb{G}\).
Proof:
Given that \(\pmb{(G,\cdot)}\) is a cyclic group.
Let \(\pmb{H}\) be a subgroup of \(\pmb{G}\).
Let \(\pmb{G=\{a^{n}:n\in \mathcal{Z} \}}\) then \(\pmb{H=\{(a^{p})^{m}:m\in \mathcal{Z} \}}\) since \(\pmb{G}\) is a cyclic and every subgroup of a cyclic group is cyclic.
To prove \(\pmb{H}\) is a characteristic subgroup of \(\pmb{G}\).
Let \(\pmb{\phi\in Aut(G)}\)
To prove \(\pmb{\phi(H)\subseteq H}\)
Let \(\pmb{x\in \phi(H)}\)
Then \(\pmb{\exists~ y\in H}\) such that \(\pmb{\phi(y)=x}\).
Let \(\pmb{\exists~ q\in \mathcal{Z} }\) such that \(\pmb{y=(a^{p})^{q}}\).
\(\implies \pmb{y=(a^{q})^{p}}\).
\(\implies \pmb{x=((\phi(a))^{q})^{p} }\) since \(\pmb{\phi}\) is a homomorphism.
Again since \(\pmb{\phi(a)\in G }\) implies \(\pmb{(\phi(a))^{q}\in G }\) then \(\pmb{\exists~ r\in \mathcal{Z} }\) such that \(\pmb{(\phi(a))^{q}=a^{r}}\).
Implies \(\pmb{x=(a^{r})^{p} }\)
\(\implies \pmb{x=(a^{p})^{r} }\)
\(\implies \pmb{x\in H}\)
Therefore \(\pmb{\phi(H)\subseteq H}\).
Hence \(\pmb{H}\) is a characteristic subgroup of \(\pmb{G}\).
Applications
Characteristic subgroups are not limited to theoretical mathematics. Their properties find applications in diverse fields such as physics where symmetry and conservation laws play crucial roles. They are also used in numerical methods and economics for studying structured systems. Additionally, their role in cryptography and coding theory is notable, where characteristic subgroups help in designing robust security frameworks and efficient data encoding schemes.
Conclusion
In conclusion, characteristic subgroups serve as a fundamental concept within group theory and broader mathematical studies. For students and researchers, understanding this topic is pivotal for progressing in higher-level mathematics, especially in courses like Abstract Algebra and various mathematical examinations. The significance of characteristic subgroups extends beyond theory, impacting various scientific and practical fields.
References
- Algebra by Michael Artin
- Abstract Algebra by David S. Dummit and Richard M. Foote
- Research Paper on Automorphism Groups
- A Course in Group Theory by John F. Humphreys
- Algebraic Structures by Richard G. Swan
- Mathematics Stack Exchange on Automorphisms
- Journal of Algebra on Inner Automorphisms
- Group Theory by Joseph J. Rotman
- Isomorphisms and Groups – Springer Link
- Automorphisms and Group Actions by László Lovász
Related Articles
- Normal Subgroups and their Properties
- Homomorphism
- Automorphism
- Inner automorphism
FAQs
- What is a characteristic subgroup? – A subgroup invariant under all automorphisms of the parent group.
- How does a characteristic subgroup differ from a normal subgroup? – Every characteristic subgroup is normal, but not all normal subgroups are characteristic.
- What is the importance of characteristic subgroups? – They help understand intrinsic properties of group structure.
- How are characteristic subgroups used in abstract algebra? – They are foundational in analyzing group symmetry and automorphisms.
- Can a characteristic subgroup be the whole group? – Yes, the entire group is always characteristic in itself.
- Are all trivial subgroups characteristic? – Yes, the trivial subgroup is always characteristic.
- Who developed the concept of characteristic subgroups? – The concept evolved from the studies of mathematicians like Galois and Klein.
- Where can characteristic subgroups be applied outside of mathematics? – In physics, economics, cryptography, and more.
- How do characteristic subgroups relate to automorphisms? – They are invariant under any automorphism of the group.
- What are some related concepts in group theory? – Normal subgroups, inner automorphisms, and group homomorphisms.
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