solvable group
A solvable group is one you can build up from abelian pieces alone — no genuinely nonabelian simple ingredient is ever needed. The name comes from Galois: a polynomial equation is solvable by radicals exactly when its Galois group is solvable, so solvability of a group is literally what tells you whether you can write the roots using nothing but field operations and nth roots.
Formally, a group G is solvable if it has a subnormal series 1 = G_0 ⊴ G_1 ⊴ ... ⊴ G_n = G with every quotient G_{i+1}/G_i abelian. Equivalently, the derived series — G, then G' = [G, G], then G'' = [G', G'], and so on — reaches the trivial subgroup in finitely many steps. For finite groups, yet another equivalent condition is that all composition factors are cyclic of prime order.
Solvable groups are closed under subgroups, quotients, and extensions, which makes them a robust class. The watershed example is the symmetric group: S_n is solvable for n ≤ 4 but not for n ≥ 5, because A_5 is a nonabelian simple group. This single fact, through Galois theory, is exactly why the general quintic cannot be solved by radicals while degrees up to four can.
Every finite p-group is solvable, since its center is nontrivial and the quotient by the center is a smaller p-group, giving an abelian-quotient chain by induction. The symmetric group S_4 is solvable via 1 ⊴ V_4 ⊴ A_4 ⊴ S_4 with abelian quotients of orders 4, 3, 2.
Solvability of p-groups and of S_4.
Two named landmarks frame the finite theory: the Feit-Thompson theorem (every finite group of odd order is solvable) and Burnside's p^a q^b theorem (every finite group whose order has only two prime divisors is solvable). Both lie far deeper than the elementary definition suggests.