Solvable Groups

Abstract Algebra · Axiom Academy

Understanding which groups can be "solved" through composition series with abelian quotients—and why the boundary lies at n=5 Think of it as building up the group in layers, where each new layer adds structure in a "simple" (abelian) way. The group can be "solved" by understanding these abelian building blocks. Let's see a concrete composition series for S₃, the symmetric group on 3 elements: A₃/ e ≅ A₃ ≅ ℤ₃ (abelian, cyclic of order 3) S₃/A₃ ≅ ℤ₂ (abelian, cyclic of order 2) Both quotients are abelian (in fact, cyclic), so S₃ is solvable! 3. The Critical Boundary: n = 5 Here's the remarkable fact: S₄ is solvable, but S₅ is not. The dividing line occurs precisely at n = 5. The alternating group A₅ has a special property: it is simple (no nontrivial normal subgroups) and non-abelian . This makes it impossible to build a composition series with abelian quotients. A₅ is the smallest non-abelian simple group A₅ appears in any composition series of S₅ Since A₅ is simple and non-abelian, it cannot be "broken down" into abelian pieces The solvability of groups is not just an abstract curiosity—it's the key to understanding when polynomial equations can be solved by radicals. Quadratic, cubic, and quartic equations have Galois groups that are subgroups of S₂, S₃, S₄—all solvable General quintic equations have Galois group S₅—not solvable This is why there's no "quintic formula" like the quadratic formula!

This is the written version of the interactive lesson above. See the full Abstract Algebra course.