Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Solvable group
Summary
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.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.