Solvable groups are a crucial concept in group theory, bridging the gap between simple and complex group structures. They're defined by a special series of subgroups, where each step leads to an abelian group.
Understanding solvable groups helps us grasp the structure of many common groups. From to , gives us insights into their properties and behavior. This knowledge is key to tackling more advanced group theory problems.
Solvable Groups: Definition and Examples
Defining Solvable Groups
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Solvable group G has subnormal series where each becomes abelian
Subnormal series for solvable group G takes form {e}=H0◃H1◃H2◃...◃Hn=G
Each Hi+1/Hi becomes abelian in this series
All become solvable
Trivial subnormal series with one abelian factor group demonstrates this property
Examples of Solvable Groups
Finite p-groups (groups of order pn where p becomes prime) always become solvable
Symmetric group Sn becomes solvable if and only if n ≤ 4
S3 and S4 become solvable
S5 and higher become non-solvable
Alternating group An becomes solvable if and only if n ≤ 4
A3 and A4 become solvable
A5 and higher become non-solvable
Dihedral group Dn always becomes solvable, regardless of n value
D3 (symmetries of equilateral triangle) becomes solvable
D4 (symmetries of square) becomes solvable
Characterizing Solvable Groups
Derived Series and Solvability
of group G defined as G(0)=G,G(1)=[G,G],G(2)=[G(1),G(1)], and so on
[A,B] denotes commutator subgroup
Group G becomes solvable if and only if derived series terminates at trivial subgroup {e} after finitely many steps
Solvable group has k where G(k)={e}
Derived length of solvable group becomes smallest k for which G(k)={e}
Measures "how many steps" needed to reach trivial subgroup
Properties of Derived Series
Factor groups G(i)/G(i+1) in derived series always become abelian
Demonstrates connection between derived series and solvability definition
Derived series provides alternative characterization of solvability
Often becomes easier to work with than subnormal series definition
Computation of derived series can determine solvability
If series terminates at {e}, group becomes solvable
If series does not terminate, group becomes non-solvable
Properties of Solvable Groups
Theorems on Subgroups and Quotients
Every subgroup of solvable group becomes solvable
Preserves solvability property in smaller structures
If G becomes solvable and N becomes of G, then G/N becomes solvable
Quotient group inherits solvability from original group
If N becomes normal subgroup of G and both N and G/N become solvable, then G becomes solvable
Solvability of normal subgroup and quotient implies solvability of whole group
Theorems on Group Operations and Orders
Direct product of solvable groups becomes solvable
Allows construction of larger solvable groups from smaller ones
Every group of order paqb, where p and q become distinct primes and a and b become non-negative integers, becomes solvable
Generalizes solvability of p-groups to groups with two prime factors
If G becomes finite group and H becomes subgroup of G with [G:H] = n, then G(n)≤H
Relates index of subgroup to derived series of group
Proof Techniques
Induction often used to prove properties of derived series
Base case: Show property holds for G(0) or G(1)
Inductive step: Assume property holds for G(k), prove for G(k+1)
Contradiction used to prove non-solvability
Assume group becomes solvable, derive contradiction to known properties
Manipulation of derived series or subnormal series
Construct series for subgroups or quotient groups
Show series terminates or continues based on original group's properties
Determining Solvability of Groups
Direct Methods
Check if group becomes abelian (all abelian groups become solvable)
Example: Cyclic group Zn always becomes solvable
For finite groups, factor order and use theorems about solvable orders
Group of order 12 = 22×3 becomes solvable
Compute derived series and check if it terminates at trivial subgroup
Example: Derived series of S3: S3>A3>{e} (solvable)
Indirect Methods
Use theorems about solvability of quotients and products for complex groups
If G/N and N become solvable, G becomes solvable
Apply knowledge of specific group families
Sn and An become solvable if and only if n ≤ 4
For matrix groups, consider Lie-Kolchin theorem
Groups of upper triangular matrices over algebraically closed field become solvable
Analyze presentations to construct subnormal series
Example: Dihedral group D4=⟨r,s∣r4=s2=1,srs=r−1⟩
Series: {e}◃⟨r2⟩◃⟨r⟩◃D4 shows solvability
Key Terms to Review (18)
Abelian groups: Abelian groups are a type of group in which the operation is commutative, meaning that for any two elements a and b in the group, the equation a * b = b * a holds true. This property makes abelian groups particularly important in various areas of mathematics, as they simplify many concepts and results in group theory. Many structures in algebra can be analyzed using the framework of abelian groups, making them a foundational concept.
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.
Composition series: A composition series is a finite sequence of subgroups of a given group, where each subgroup is normal in the next one, and the factor groups formed by consecutive subgroups are simple groups. This concept is significant as it provides insight into the structure of groups, particularly when analyzing solvable groups and their properties, as well as in the context of Galois theory, which connects group theory to field extensions.
Conjugacy Class: A conjugacy class in group theory is a set of elements in a group that are related to each other by conjugation. Specifically, if an element 'g' can be transformed into another element 'h' by an inner automorphism, i.e., there exists an element 'x' in the group such that 'h = xgx^{-1}', then 'g' and 'h' belong to the same conjugacy class. This concept is crucial as it helps in understanding the structure of groups, particularly in relation to normal subgroups and representation theory.
Derived Series: The derived series of a group is a sequence of subgroups where each subgroup is generated by the commutator of the previous subgroup with itself. This series helps in analyzing the structure of a group by breaking it down into simpler pieces and determining its properties, such as solvability. It connects to key concepts like solvable groups, nilpotent groups, and their applications in fields like Galois theory.
Dihedral Groups: Dihedral groups are a class of groups that represent the symmetries of a regular polygon, including both rotations and reflections. Specifically, the dihedral group of order n, denoted as D_n, has n rotations and n reflections, making it a key example in group theory. These groups play an important role in understanding symmetries, structures, and properties related to p-groups and solvable groups.
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.
Feit-Thompson Theorem: The Feit-Thompson Theorem states that every finite group of odd order is solvable. This theorem plays a crucial role in the classification of finite groups and highlights the relationship between group order and solvability, providing insight into the structure of groups and their properties.
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.
John Thompson: John Thompson was a prominent mathematician known for his work in group theory, particularly in the study of solvable groups and their properties. His contributions laid foundational groundwork for understanding the structure and classification of solvable groups, which are critical in the broader context of abstract algebra and its applications.
Nilpotent Groups: Nilpotent groups are a class of groups where the upper central series terminates in the group itself. This means that every normal subgroup of the group has a non-trivial intersection with the center of the group. These groups are important because they exhibit certain properties that connect to solvable groups, as all nilpotent groups are also solvable, but not all solvable groups are nilpotent. They play a significant role in understanding the structure and classification of groups.
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-groups: A p-group is a group whose order (the number of elements in the group) is a power of a prime number p. This concept is essential in understanding group theory, particularly in the context of analyzing the structure and properties of finite groups. P-groups exhibit unique characteristics, such as having nontrivial centers and being nilpotent, which make them a fundamental part of group classification and the study of solvable groups.
Simple Groups: Simple groups are nontrivial groups that do not have any normal subgroups other than the trivial group and the group itself. These groups serve as the building blocks for all finite groups, as any finite group can be expressed as a combination of simple groups through various structures, making them essential in understanding group theory.
Solvability: Solvability refers to the property of a group where it can be broken down into simpler components, specifically, a group that has a series of subgroups where each factor group is abelian. This concept is crucial as it helps in understanding the structure of groups and provides insights into their classification. Solvable groups are linked to various properties such as their representation, their behavior under certain operations, and their relation to more complex groups.
Subgroup series: A subgroup series is a finite sequence of subgroups of a group where each subgroup is a normal subgroup of the next one in the series. This concept is crucial for understanding the structure of groups, particularly in the study of solvable groups, where each factor group formed by consecutive subgroups plays an important role in determining the group's solvability.
Symmetric groups on 4 elements: The symmetric group on 4 elements, denoted as S4, is the group consisting of all the permutations of a set with 4 distinct elements. It has a total of 24 elements, which corresponds to the number of ways to arrange 4 objects, showcasing various properties such as group operations and structure, making it a central object of study in group theory, particularly when analyzing solvable groups and their characteristics.
William Burnside: William Burnside was a prominent mathematician known for his contributions to group theory and combinatorial enumeration. His work established foundational principles such as Burnside's Lemma, which relates to counting distinct configurations under group actions, helping bridge the study of groups and geometric structures.