Understanding the Normalizer Subgroup in Group Theory
Welcome to our detailed lesson on the normalizer subgroup in group theory. In this article, we explore the concept of the normalizer of a subgroup along with related ideas like invariance under an element. You will find clear definitions, engaging examples, and rigorous proofs of key theorems. This guide uses active voice and simple language to ensure that both beginners and advanced learners can easily understand the content.
Invariant Under an Element
Definition:
Let \( (G, \circ) \) be a group and let \( H \) be a subgroup of \( G \), with \( a \in G \). We say that \( H \) is invariant under \( a \) if
\[ aHa^{-1} = H. \]
In other words, conjugating \( H \) by \( a \) leaves \( H \) unchanged. This concept is crucial as it lays the groundwork for understanding subgroup normalizers.
Normalizer of a Subgroup
Definition:
Let \( (G, \circ) \) be a group and let \( H \) and \( K \) be subgroups of \( G \). The normalizer subgroup of \( H \) in \( K \), denoted by \( N_{K}(H) \), is defined as:
\[ N_{K}(H) = \{ k \in K : kHk^{-1} = H \}. \]
Essentially, \( N_{K}(H) \) is the set of all elements in \( K \) that, when conjugating \( H \), leave it invariant. This measurement tells us how “stable” \( H \) is within \( K \), and it plays an important role in understanding the structure of groups.
Example: Normalizer in the Symmetric Group
Consider the symmetric group \( S_3 \). Let \( H = \{ e, (1\,2) \} \) be a subgroup of \( S_3 \), and set \( K = S_3 \). The normalizer \( N_{S_3}(H) \) comprises all elements \( k \in S_3 \) such that:
\[ kHk^{-1} = H. \]
This example shows how the normalizer subgroup captures those elements in \( S_3 \) that stabilize \( H \) under conjugation, thereby maintaining its structure.
Theorems and Proofs
Theorem 1: \( N_{K}(H) \) is a Subgroup of \( K \)
Statement:
Let \( (G, \circ) \) be a group and \( H \) and \( K \) be subgroups of \( G \). Then the normalizer \( N_{K}(H) \) is itself a subgroup of \( K \).
Proof: We prove this by verifying the subgroup criteria:
- Non-emptiness: The identity element \( e \in K \) satisfies \[ eHe^{-1} = H, \] hence \( e \in N_{K}(H) \).
- Closure: For any \( k_1, k_2 \in N_{K}(H) \), we have \[ (k_1 \circ k_2)H(k_1 \circ k_2)^{-1} = k_1\big(k_2 H k_2^{-1}\big)k_1^{-1} = k_1 H k_1^{-1} = H. \] Thus, \( k_1 \circ k_2 \in N_{K}(H) \).
- Inverses: For any \( k \in N_{K}(H) \), since \[ kHk^{-1} = H, \] taking inverses gives \[ k^{-1} H k = H, \] which implies \( k^{-1} \in N_{K}(H) \).
Therefore, \( N_{K}(H) \) satisfies all subgroup properties and is indeed a subgroup of \( K \).
Theorem 2: The Number of Distinct Conjugates Equals the Coset Index
Statement:
Let \( (G, \circ) \) be a group and \( H \) and \( K \) be subgroups of \( G \). Then the number of distinct conjugates of \( H \) induced by elements of \( K \) equals the index \( [K : N_{K}(H)] \).
Proof: Define the set of distinct conjugates as
\[
T = \{ kHk^{-1} : k \in K \},
\]
and the set of left cosets of \( N_{K}(H) \) in \( K \) as
\[
S = \{ kN_{K}(H) : k \in K \}.
\]
We then construct a mapping \( f: T \to S \) by
\[
f(kHk^{-1}) = kN_{K}(H).
\]
Well-Defined: If \( aHa^{-1} = bHb^{-1} \) for \( a, b \in K \), then \( b^{-1}a \in N_{K}(H) \), which implies \( aN_{K}(H) = bN_{K}(H) \). Hence, \( f(aHa^{-1}) = f(bHb^{-1}) \).
Injectivity: If \( f(aHa^{-1}) = f(bHb^{-1}) \), then \( aN_{K}(H) = bN_{K}(H) \). This yields \( b^{-1}a \in N_{K}(H) \) and therefore \( aHa^{-1} = bHb^{-1} \).
Surjectivity: Every coset \( kN_{K}(H) \in S \) is the image of \( kHk^{-1} \in T \).
Since \( f \) is bijective, the number of distinct conjugates of \( H \) is equal to \( [K : N_{K}(H)] \).
Theorem 3: Conjugate Count for Finite Subgroups
Statement:
Let \( (G, \circ) \) be a group and let \( H \) and \( K \) be two finite subgroups of \( G \). If \( H \) is invariant under \( n \) elements of \( K \), then \( H \) has \(\frac{|K|}{n}\) distinct conjugates by elements of \( K \).
Proof: Assume that \[ |N_{K}(H)| = n. \] By Theorem 2, the number of distinct conjugates of \( H \) is given by the index \[ [K : N_{K}(H)]. \] According to Lagrange’s theorem, \[ |K| = [K : N_{K}(H)] \cdot |N_{K}(H)|, \] which simplifies to \[ [K : N_{K}(H)] = \frac{|K|}{n}. \] This confirms that the number of distinct conjugates of \( H \) in \( K \) is exactly \(\frac{|K|}{n}\).
In conclusion, understanding the normalizer subgroup is key to grasping the internal structure and symmetry of groups. Through clear definitions, illustrative examples, and rigorous proofs, we see how normalizers help determine the stability of subgroups under conjugation. This foundational concept not only deepens our insight into group theory but also enhances our appreciation of algebraic structures.
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FAQs
Group Theory
- What is a group in group theory?
A group is a set equipped with a binary operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverses for every element.
- What are the main properties of a group?
The four main properties are:
- Closure: The result of the operation on any two elements of the group is also in the group.
- Associativity: The group operation is associative.
- Identity: There exists an element that does not change other elements when used in the operation.
- Invertibility: Every element has an inverse that, when combined with the element, yields the identity.
- What is the identity element in a group?
The identity element is a unique element in the group that, when combined with any other element using the group operation, leaves that element unchanged. It is commonly denoted by e or 1.
- What is an abelian group?
An abelian group is one in which the binary operation is commutative. This means for any two elements a and b in the group, a · b = b · a
- What is a subgroup?
A subgroup is a subset of a group that is itself a group under the same binary operation. It must satisfy the group properties: closure, associativity, identity, and inverses.
- What is a normal subgroup and how does it relate to quotient groups?
A normal subgroup is a subgroup that is invariant under conjugation by any element of the original group. This means for every element n in the normal subgroup N and every element g in the group, gng⁻¹ is still in N. Normal subgroups allow the construction of quotient groups, where the group is partitioned into cosets of the normal subgroup.
- What are group homomorphisms?
A group homomorphism is a function between two groups that preserves the group operation. This means if f: G → H is a homomorphism and a, b are elements of G, then f(a · b) = f(a) · f(b) in H.
- What is Lagrange’s theorem in group theory?
Lagrange's theorem states that for any finite group, the order (number of elements) of every subgroup divides the order of the entire group. This theorem is a fundamental result in the study of finite groups.
- What is Cayley’s theorem?
Cayley’s theorem states that every group is isomorphic to a subgroup of a symmetric group. This implies that every group can be represented as a group of permutations acting on a set.
- How is group theory applied in other fields?
Group theory has applications in many fields including:
- Physics: Describing symmetries and conservation laws.
- Chemistry: Analyzing molecular symmetry and chemical bonding.
- Cryptography: Underlying structures in cryptographic systems.
- Mathematics: Foundational in algebra, geometry, and number theory.
