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In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable (note this theorem holds only in characteristic 0). This means associated to a polynomial there is a tower of field extensionssuch that where , so is a solution to the equation where contains a splitting field for For example, the smallest Galois field extension of containing the elementgives a solvable group. It has associated field extensionsgiving a solvable group of Galois extensions containing the following composition factors: with group action , and minimal polynomial . with group action , and minimal polynomial . with group action , and minimal polynomial containing the 5th roots of unity excluding . with group action , and minimal polynomial . where is the identity permutation. All of the defining group actions change a single extension while keeping all of the other extensions fixed. For example, an element of this group is the group action . A general element in the group can be written as for a total of 80 elements. It is worthwhile to note that this group is not abelian itself. For example: In fact, in this group, . The solvable group is isometric to , defined using the semidirect product and direct product of the cyclic groups. In the solvable group, is not a normal subgroup. A group G is called solvable if it has a subnormal series whose factor groups (quotient groups) are all abelian, that is, if there are subgroups 1 = G0 < G1 < ⋅⋅⋅ < Gk = G such that Gj−1 is normal in Gj, and Gj /Gj−1 is an abelian group, for j = 1, 2, ..., k.

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