Soluble Groups in Algebra: What Are They and What Branch Do They Fall Under?

Click For Summary
Soluble groups, also known as solvable groups, are a concept in algebra related to the structure of groups and their composition series. The terms "soluble" and "solvable" may reflect regional language differences, with "solvable" being more common in American English and "soluble" in British English. The discussion references Galois' results, indicating that a polynomial equation is solvable by radicals if the corresponding symmetric group is solvable, particularly for degrees less than five. Historical usage of the terms is noted, with "soluble" appearing in the 1902 Encyclopedia Britannica and "solvable" in E. Netto's work from 1892. Understanding these terms is essential for grasping the foundational concepts in group theory within algebra.
pivoxa15
Messages
2,250
Reaction score
1
How does soluble groups fit into algebra? Why is there another name for it called solvable groups?

What branch does it fall under?
 
Physics news on Phys.org
Maybe "solvable" is American and "soluble" is English. For example, "formula" is American and "formulae" is English. Look at the author's name of the book, is it English?
 
That is hard to swallow. I've only heard soluble made reference as a chemistry term.
 
Formula is latin, and is the same in both English and American English. Formulae is the 'proper' plural in both languages too, though formulas is now acceptable in either.
 
pivoxa15 said:
That is hard to swallow. I've only heard soluble made reference as a chemistry term.

How is this hard to swallow?

It's as easy as... English book means its solvable.
 
Do you agree with me it is just a difference in language?
 
I suppose solvable groups get their name from the results of Galois, "The general polynomial equation of degree n is solvable by radicals if and only if S_n (the symmetric group) is solvable".

(Together with the trivial observation that "S_n is solvable if and only if n is inferior to 5", this constitute an argument of the unsolvability of general polynomial equations of degree >4 by radiacals)
 
Last edited:
Especially since Galois was french!
 
  • #10
Hello all.

The Oxford English Dictionary which purports to give the first written use of all words in the English language has an entry for soluble:

Soluble (Math)=solvable. Giving its first recorded use in this sense as being in the 1902 edition of the Encyclopedia Britannica. The quote is:-

''A group defines uniquely the set of factor-groups that occur in its composition series ...When the order of all the factor groups are primes the group is said to be soluble.''

It also gives the first recorded use of the word solvable in the mathematical sense as being in 1892 by E.Netto in His Theory of Substitutions.

I hope this is of some use.

Matheinste.

Matheinste.
 

Similar threads

Insights Groups, The Path from a Simple Concept to Mysterious Results
  • · Replies 17 ·
Replies
17
Views
9K
Undergrad Differences between left vs right actions in some group theory questions
  • · Replies 1 ·
Replies
1
Views
520
Graduate Near-Rings with Noncommutative Addition and Two-Sided Distributivity
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Classification of reductive groups via root datum
  • · Replies 3 ·
Replies
3
Views
2K
High School What unique contributions to math did linear algebra make?
  • · Replies 19 ·
Replies
19
Views
4K
Undergrad Dfns group action & group, written in terms of the other
  • · Replies 26 ·
Replies
26
Views
846
Insights Lie Algebras: A Walkthrough - The Structures
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Meaning of "passage to the quotient"?
  • · Replies 3 ·
Replies
3
Views
871
Insights Lie Algebras: A Walkthrough - The Basics
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Question about "A Group Epimorphism is Surjective"
  • · Replies 13 ·
Replies
13
Views
1K