Solvable group . In abstract algebra, especially in group theory, this is the name of a group that is a member of a succession of subgroups that respect a partial order and are useful in Galois Theory and in the analysis of solving algebraic equations by formulas involving radicals and operations with the respective coefficients.
Let be the sequence of normal subgroups
G → G (1) → G (2) → … → G (k) → G (k + 1) → … (2)
Group G is called resolvable , if the sequence (2) stops at the unit subgroup, that is G (m) = e for a certain minimum index of m degrees of solubility of group G. It is notable that every abelian group, among these also cyclical, it is resolvable grade 1. [1]
Summary
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- 1 switch
- 1 Switching
- 2 Proposition
- 2 References
- 3 Sources
- 4 Linked pages
Switch
The expression [x, y] = xyx -1 y -1 ,
It is called the commutator of the elements x, and of the group G, it is used as a corrective term, when there is a need to alter the locations of x and y:
xy = [x, y] xy
When x and y are interchangeable, it turns out that the switch is e; [x, y] = e
Switching
Let L be the set of all commutators in G. The subgroup G ‘ is called the commutator (or derived subgroup ) of group G: = G (1) = [G, G] generated by the set L.
G ‘= <[x, y] / x, and they are in G>.
Although the inverse of a switch is a switch, the product of two switches is not always a switch. [2]
Proposition
Any subgroup K of G, which contains the commutator G ‘of group G is normal with respect to G. The quotient group G / G’ is abelian and G ‘is contained in each normal subgroup K, such that G / K is abelian ( the maximum order of the abelian quotient group G / K is equal to the index of G: G ‘).
The resolubility of group S 4 and all its subgroups is the ratio of radical solubility of algebraic equations of degree no greater than 4.
