Solvable GroupEdit
Solvable groups form a core concept in abstract algebra, capturing the idea that a potentially complicated symmetry structure can be built up from simpler, commutative pieces. The key device is the derived series, a chain of subgroups whose successive quotients measure how far the group is from being commutative. A finite group is solvable if this process terminates in the trivial subgroup after finitely many steps. Since abelian groups are the simplest solvable groups, the theory starts with those and explains how more intricate groups still retain an abelian-building block structure.
In the broader mathematical context, solvable groups are tightly linked to the solvability of polynomial equations by radicals. The nineteenth-century development of Galois theory shows that a polynomial’s roots can be expressed by radicals precisely when its Galois group is solvable. This ties a purely algebraic property of a group to a constructive question about solving equations. The interplay between solvable and non-solvable groups also helps explain why certain equations, such as the general quintic, resist explicit formulae in radicals.
Definitions
- Derived series: For a group G, define G^(0) = G and G^(i+1) = [G^(i), G^(i)], the commutator subgroup of G^(i). A group is solvable if G^(n) = {e} for some n, i.e., after finitely many steps the series reaches the trivial group. See Commutator subgroup and Derived series.
- Alternative formulation: A group is solvable if there exists a finite chain of subgroups {e} = G_0 ⊆ G_1 ⊆ ... ⊆ G_n = G such that each G_{i-1} is normal in G_i and the quotient G_i/G_{i-1} is abelian. This mirrors the idea of building G by successive abelian extensions.
- Solvable length: The smallest n with G^(n) = {e} is called the solvable length of G.
These notions have several clarifying consequences. Finite p-groups (groups of order p^k for a prime p) are solvable. If N is solvable and G/N is solvable, then G is solvable (solvability is preserved under extensions). Every subgroup and every quotient of a solvable group is solvable, so solvability passes to many constructions built from solvable pieces. A finite group is solvable if and only if all its composition factors are cyclic of prime order.
Examples
- Abelian groups: Every abelian group is solvable, since the commutator subgroup is trivial. Examples include cyclic groups Cyclic group and products of cyclic groups.
- Small non-abelian solvable groups: The symmetric group on three letters, S_3, is solvable because its derived subgroup is a cyclic group of order 3 and its next derived subgroup is trivial. The dihedral groups Dihedral group are solvable as well.
- Symmetric groups: S_n is solvable for n ≤ 4 but not for n ≥ 5. In particular, S_5 is not solvable, a fact tied to the simplicity of A_5 and its non-abelian nature; the derived chain cannot reach the identity.
- Alternating group A_5: A_5 is simple and non-abelian, hence not solvable. This example is central to the historical connection between group theory and the limits of solving polynomials by radicals.
- p-groups and nilpotent groups: Finite p-groups are solvable and play a key role in the structure theory of solvable groups. Nilpotent groups, a subclass with stronger conditions, are always solvable as well.
The role in Galois theory and polynomial equations
- Abel–Ruffini connection: The question of whether general polynomials of degree five or higher can be solved by radicals is answered negatively in general because their Galois groups are, in the worst case, isomorphic to S_n with n ≥ 5, which is not solvable. See Abel–Ruffini theorem.
- Solvability as a test for solvable polynomials: When a polynomial’s Galois group is solvable, its roots can often be expressed by radicals, or by a sequence of radical expressions together with known algebraic operations. This is the bridge between abstract group structure and concrete formulas.
- Classical breakthroughs: Burnside’s pq-theorem (finite groups of order p^a q^b are solvable) and the Feit–Thompson theorem (every finite group of odd order is solvable) are landmark results that deepen the understanding of how solvable groups sit inside the landscape of all finite groups. See Burnside's theorem and Feit–Thompson theorem.
Structural aspects and consequences
- Composition factors: A finite solvable group has a composition series with abelian factors, hence cyclic of prime order. This factorization into simple abelian pieces helps explain why these groups are, in a sense, built from commutative blocks.
- Extensions and normal structure: Many solvable groups arise from sequences of abelian extensions. Normal series with abelian quotients illuminate how complexity can be layered and controlled.
- Relation to non-solvable groups: The existence of non-solvable groups (e.g., S_n for n ≥ 5 or A_5) provides a sharp boundary between solvable and non-solvable symmetry structures. This contrast is central to understanding why some equations resist closed-form solutions.