8.1 Commutators and Derived Series
4 min read•august 16, 2024
Commutators and derived series are powerful tools for understanding group structure. They measure non-commutativity and help identify important subgroups, shedding light on a group's internal workings and relationships between elements.
These concepts are crucial for studying solvable groups, a key focus of this unit. By examining how commutators behave and how the derived series progresses, we can determine if a group is solvable and explore its composition.
Commutators in Group Theory
Definition and Properties of Commutators
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- of elements a and b in group G defined as
- Measures extent of non-commutativity between a and b
- Commutator subgroup [G,G] generated by all commutators in G
- of G
- Trivial [G,G] implies G is abelian (cyclic groups)
- Non-trivial [G,G] indicates non- (dihedral groups)
- Key identities for commutators
- Perfect groups satisfy G = [G,G] (special linear group SL(2,5))
- Simple groups have no proper normal subgroups (alternating group A5)
Role of Commutators in Group Theory
- Essential for measuring group non-commutativity
- Crucial in studying normal subgroups and quotient groups
- Used to define and analyze perfect and simple groups
- Applied in representation theory to study group actions
- Fundamental in the study of Lie algebras and Lie groups
- Employed in the analysis of group extensions and central products
- Utilized in the investigation of group automorphisms and isomorphisms
Derived Series of a Group
Construction and Properties
- Derived series defined recursively
- Each term in derived series characteristic subgroup of G
- Descending chain of normal subgroups
- Solvable groups have derived series terminating at trivial subgroup (symmetric group S4)
- Derived length finite number of steps to reach trivial subgroup
- Quotient group G/G^(1) always abelian
- Largest abelian quotient of G
Significance in Group Structure Analysis
- Provides crucial information about group solvability
- Used to determine simplicity of groups
- Helps identify composition series of a group
- Employed in the study of group extensions
- Useful in analyzing Sylow subgroups of finite groups
- Applied in Galois theory to investigate field extensions
- Facilitates understanding of group representations
Theorems of Commutators and Derived Series
Fundamental Theorems
- Commutator subgroup [G,G] smallest normal subgroup N with G/N abelian
- For normal subgroups H and K of G, [H,K] normal subgroup contained in H ∩ K
- Three Subgroups Lemma
- For subgroups H, K, L of G
- If [H,K,L] = [K,L,H] = 1, then [L,H,K] = 1
- Finite group G solvable if and only if subnormal series with abelian factors exists
- Subgroups and quotient groups of solvable groups are solvable
- Direct product of solvable groups is solvable
- In finite group G, subgroup of index p (smallest prime dividing |G|) is normal
Proofs and Demonstrations
- Prove commutator subgroup theorem using normal closure properties
- Demonstrate [H,K] normality using commutator identities and subgroup conjugation
- Establish Three Subgroups Lemma through repeated application of commutator identities
- Show solvability equivalence using induction on group order
- Prove solvability of subgroups and quotients using derived series properties
- Demonstrate solvability of direct products by constructing combined derived series
- Establish normality of index p subgroup using Sylow's theorems and commutator calculations
Applications of Commutators and Derived Series
Problem-Solving Techniques
- Determine group abelian nature by calculating commutators
- Abelian if all commutators are trivial (cyclic groups)
- Non-abelian if non-trivial commutators exist (quaternion group)
- Calculate commutator subgroups for various group types
- Dihedral groups: [D_n, D_n] = rotations of order n/2 or n (even or odd n)
- Symmetric groups: [S_n, S_n] = A_n for n ≥ 2
- Matrix groups: SL(n,F) for SL(n,F) with n ≥ 2 and |F| > 3
- Construct derived series to determine group solvability
- Solvable: symmetric group S4
- Non-solvable: alternating group A5
- Analyze semidirect and wreath product structures
- Semidirect product: Z_3 ⋊ Z_2
- Wreath product: Z_2 ≀ Z_3
Advanced Applications
- Study p-groups and nilpotent groups using commutators
- Upper central series: Z_1(G) ⊆ Z_2(G) ⊆ ...
- Lower central series: G = γ_1(G) ⊇ γ_2(G) ⊇ ...
- Classify small order groups with commutator calculations
- Groups of order 8: 2 abelian, 3 non-abelian
- Groups of order 12: 2 abelian, 3 non-abelian
- Investigate Galois group solvability in field theory
- Solvable: cyclotomic extensions
- Non-solvable: general polynomial of degree ≥ 5
- Apply to representation theory and character theory
- Determine irreducible representations using
- Analyze character degrees using commutator subgroup
Key Terms to Review (16)
Abelian group: An abelian group is a type of group where the group operation is commutative, meaning that for any two elements in the group, the result of the operation does not depend on the order in which they are combined. This property leads to many important results and applications across various areas in group theory and beyond.
Abelianization: Abelianization is the process of transforming a group into its abelian (commutative) form by factoring out its commutator subgroup. This operation essentially measures how far a group is from being abelian by capturing all the 'non-commutative' behavior within the group. The resulting quotient group, which is called the abelianization of the original group, allows for a clearer understanding of the structure and properties of the group in terms of abelian characteristics.
Burnside's Theorem: Burnside's Theorem states that the number of distinct orbits of a finite group acting on a set can be computed using the average number of points fixed by each group element. This theorem is crucial in understanding how groups can act on sets, providing insights into symmetries and combinatorial structures, particularly within the study of commutators, derived series, solvable groups, and nilpotent groups.
Commutator: A commutator is a mathematical construct in group theory, defined for two elements a and b of a group G as the element [a, b] = a^{-1}b^{-1}ab. This operation measures how much the two elements fail to commute. If the commutator is the identity element of the group, then the elements a and b commute, which connects to broader concepts like normal subgroups and derived series.
Derived subgroup: A derived subgroup, also known as the commutator subgroup, is the subgroup generated by all commutators of a group. It serves as a measure of how non-abelian a group is, since the derived subgroup captures the 'failure' of the group to be abelian. This concept is crucial for understanding the structure of groups and analyzing their properties, especially in relation to the derived series, which examines a sequence of derived subgroups to study a group's solvability.
Factor Group: A factor group, also known as a quotient group, is a type of group formed by partitioning a group into disjoint subsets called cosets of a normal subgroup. This structure allows for the creation of a new group that retains properties of the original group while simplifying its structure. Factor groups are essential in understanding the relationships between groups, especially in the context of normal subgroups, which play a key role in their formation.
Group Homomorphism: A group homomorphism is a function between two groups that preserves the group operation, meaning if you take any two elements from the first group, their images in the second group will combine in the same way as they did in the first group. This concept is crucial for understanding how different groups relate to each other, and it connects deeply with properties such as normal subgroups, quotient groups, and various structural aspects of groups.
Inner automorphism: An inner automorphism is a specific type of automorphism of a group defined by conjugation with an element of the group itself. For any element 'g' in a group 'G', the inner automorphism induced by 'g' is the function that takes any element 'x' in 'G' to 'gxg^{-1}'. This concept highlights the structural properties of groups and how elements relate to each other through symmetry and transformation, which is essential in understanding the overall structure of the group.
Jordan-Hölder Theorem: The Jordan-Hölder Theorem states that in a finite group, any two composition series of the group have the same length and their factors, known as simple factors, are isomorphic up to order. This theorem underlines the importance of the structure of groups, showcasing how a group's composition can be understood through its simple factors, which are crucial in analyzing group properties.
Length of a Series: The length of a series refers to the number of steps or levels in a derived series formed from a group, typically based on commutators. In the context of group theory, this concept helps in understanding how far one must go through successive derived subgroups to reach the trivial group, shedding light on the group's structure and its properties related to solvability.
Nilpotent group: A nilpotent group is a type of group where the upper central series terminates at the group itself after a finite number of steps. This means that every non-trivial normal subgroup of the group has a non-trivial intersection with the center, which allows for the group's structure to be decomposed in a way that reveals its underlying simplicity. This concept connects deeply with the behavior of commutators and derived series, highlighting the intrinsic relationship between nilpotent groups and their properties, as well as their applications in various areas of mathematics.
Normal Subgroup: A normal subgroup is a subgroup that is invariant under conjugation by any element of the group, meaning that for a subgroup H of a group G, for all elements g in G and h in H, the element gHg^{-1} is still in H. This property allows for the formation of quotient groups and is essential in understanding group structure and homomorphisms.
P-group: A p-group is a group in which the order of every element is a power of a prime number p. This characteristic makes p-groups particularly interesting in group theory, as they exhibit unique structural properties and behaviors. Their connection to Sylow theorems highlights their importance in understanding the composition of finite groups and their subgroups.
Simple Group: A simple group is a nontrivial group that has no normal subgroups other than the trivial group and itself. This property makes simple groups the building blocks for all finite groups, as any finite group can be constructed from simple groups through various operations like direct products and extensions.
Solvable group: A solvable group is a type of group in abstract algebra where the derived series eventually reaches the trivial subgroup. This property indicates that the group can be broken down into simpler components, making it easier to study its structure and behavior. Solvable groups play a crucial role in various mathematical contexts, including understanding commutators, applications involving nilpotent groups, and their implications in Galois theory.
Successive quotients: Successive quotients are a series of quotient groups derived from a group and its normal subgroups, often used to analyze the structure of groups through their layers of simplicity. This concept is particularly relevant when exploring derived series, where each successive quotient provides insights into the commutator structure and the overall behavior of the group as one investigates its building blocks through abelian factors.