⭕Groups and Geometries Unit 8 – Solvable and Nilpotent Groups

Key Concepts and Definitions

• Solvable groups are a class of groups that can be constructed from abelian groups using extensions
• The derived series of a group $G$ is defined as $G = G^{(0)} \trianglerighteq G^{(1)} \trianglerighteq \cdots \trianglerighteq G^{(n)} = \{e\}$, where $G^{(i+1)} = [G^{(i)}, G^{(i)}]$
  • $[G^{(i)}, G^{(i)}]$ denotes the subgroup generated by commutators of elements in $G^{(i)}$
• A group $G$ is solvable if its derived series terminates at the trivial subgroup $\{e\}$ after a finite number of steps
• Nilpotent groups are a subclass of solvable groups with a more stringent condition on their central series
• The lower central series of a group $G$ is defined as $G = G_1 \trianglerighteq G_2 \trianglerighteq \cdots \trianglerighteq G_n = \{e\}$, where $G_{i+1} = [G, G_i]$
  • $[G, G_i]$ denotes the subgroup generated by commutators of elements in $G$ and $G_i$
• A group $G$ is nilpotent if its lower central series terminates at the trivial subgroup $\{e\}$ after a finite number of steps
• The nilpotency class of a nilpotent group is the length of its lower central series minus one

Properties of Solvable Groups

• Every abelian group is solvable, as its derived series terminates after one step
• Subgroups of solvable groups are solvable
• Quotient groups of solvable groups are solvable
• The direct product of solvable groups is solvable
• If $N$ is a normal subgroup of $G$ and both $N$ and $G/N$ are solvable, then $G$ is solvable (solvability is extension-closed)
• Finite $p$-groups (groups with order equal to a prime power) are solvable
• The symmetric group $S_n$ is solvable for $n \leq 4$, but not for $n \geq 5$
  • $S_5$ is the smallest non-solvable group

Nilpotent Groups: Basics and Examples

• Every abelian group is nilpotent, as its lower central series terminates after one step
• Subgroups of nilpotent groups are nilpotent
• Quotient groups of nilpotent groups are nilpotent
• The direct product of nilpotent groups is nilpotent
• Finite $p$-groups are nilpotent
  • The converse is not true: the quaternion group $Q_8$ is nilpotent but not a $p$-group
• The center of a nilpotent group is always non-trivial (except for the trivial group)
• Examples of nilpotent groups include:
  • The dihedral group $D_8$ (nilpotency class 2)
  • The quaternion group $Q_8$ (nilpotency class 2)
  • The Heisenberg group (nilpotency class 2)

Subgroup Structure and Normal Series

• A normal series of a group $G$ is a sequence of subgroups $G = G_0 \trianglerighteq G_1 \trianglerighteq \cdots \trianglerighteq G_n = \{e\}$ such that $G_{i+1}$ is normal in $G_i$ for all $i$
• The factors of a normal series are the quotient groups $G_i/G_{i+1}$
• A subnormal series is a generalization of a normal series where the subgroups are not required to be normal, only subnormal
• The Jordan-Hölder theorem states that any two composition series of a group have the same length and isomorphic factors (up to permutation)
• Solvable groups can be characterized by the existence of a subnormal series with abelian factors
• Nilpotent groups have a more refined subgroup structure compared to solvable groups
  • Every maximal subgroup of a nilpotent group is normal
  • Every subgroup of a nilpotent group is subnormal

Commutator Subgroups and Derived Series

• The commutator of elements $a$ and $b$ in a group $G$ is defined as $[a, b] = a^{-1}b^{-1}ab$
• The commutator subgroup (or derived subgroup) of a group $G$, denoted $G'$ or $[G, G]$, is the subgroup generated by all commutators of elements in $G$
  • $G'$ is always a normal subgroup of $G$
• The derived series of a group $G$ is obtained by repeatedly taking commutator subgroups: $G^{(0)} = G$, $G^{(i+1)} = (G^{(i)})'$
• The derived length of a solvable group is the length of its derived series
• Commutators in nilpotent groups have additional properties:
  • In a nilpotent group, the commutator operation is multi-linear and satisfies the Jacobi identity
  • The commutator subgroup of a nilpotent group is nilpotent

Nilpotency vs. Solvability: Comparisons

• Every nilpotent group is solvable, but the converse is not true
  • The smallest solvable, non-nilpotent group is the symmetric group $S_3$
• Nilpotent groups have a more rigid structure compared to solvable groups
  • Nilpotent groups have a non-trivial center, while solvable groups may not
  • Every maximal subgroup of a nilpotent group is normal, while solvable groups may have non-normal maximal subgroups
• The derived series and the lower central series coincide for nilpotent groups, but not for solvable groups
• Nilpotent groups can be characterized by the existence of a central series, while solvable groups can be characterized by the existence of a subnormal series with abelian factors
• Nilpotent groups have polynomial growth, while solvable groups can have exponential growth

Applications in Group Theory

• Solvable groups play a crucial role in the classification of finite simple groups
  • The Feit-Thompson theorem states that every finite group of odd order is solvable
  • The classification of finite simple groups shows that every non-abelian finite simple group is either a cyclic group of prime order, an alternating group $A_n$ ($n \geq 5$), a simple group of Lie type, or one of the 26 sporadic groups
• Nilpotent groups are used in the study of group cohomology and representation theory
  • The dimension subgroups of a group $G$ are defined using the lower central series and are related to the cohomology of $G$
  • Nilpotent groups have a well-behaved representation theory, with all irreducible representations being monomial
• Solvable and nilpotent groups appear in various areas of mathematics, such as algebraic topology, number theory, and algebraic geometry
  • The fundamental group of a solvmanifold (a compact manifold with a transitive solvable group action) is a solvable group
  • The class groups of certain number fields, such as quadratic fields, are often solvable or nilpotent

Problem-Solving Techniques

• To show that a group is solvable, construct a subnormal series with abelian factors or show that its derived series terminates at the trivial subgroup
  • For example, to prove that $S_4$ is solvable, consider the series $S_4 \trianglerighteq A_4 \trianglerighteq V_4 \trianglerighteq \{e\}$, where $V_4$ is the Klein four-group
• To show that a group is nilpotent, construct a central series or show that its lower central series terminates at the trivial subgroup
  • For example, to prove that $D_8$ is nilpotent, consider the lower central series $D_8 \trianglerighteq \langle r^2 \rangle \trianglerighteq \{e\}$
• Use the properties of solvable and nilpotent groups to simplify problems
  • For instance, if $G$ is nilpotent and $H$ is a proper subgroup, then $H$ is properly contained in its normalizer $N_G(H)$
• Utilize the relationship between solvability, nilpotency, and other group properties
  • For example, if a group has a non-trivial center and every maximal subgroup is normal, then it is nilpotent
• Apply the Feit-Thompson theorem or the classification of finite simple groups when dealing with finite groups of odd order or non-solvable groups
• Consider using induction on the order of the group or the length of the derived or lower central series to prove statements about solvable or nilpotent groups