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

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Discussion Overview

The discussion revolves around the concept of soluble (or solvable) groups in algebra, exploring their definitions, terminology differences, and historical context. Participants inquire about the branch of mathematics these groups belong to and the implications of their naming conventions.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Historical

Main Points Raised

  • One participant questions how soluble groups fit into algebra and why they are referred to as solvable groups.
  • Another suggests that the difference in terminology may be due to regional language variations, comparing it to other English terms.
  • Some participants express skepticism about the use of "soluble" in this context, associating it primarily with chemistry.
  • A participant references Galois' results regarding solvable groups and their relation to polynomial equations, noting a specific condition involving the symmetric group S_n.
  • There is a mention of the Oxford English Dictionary's definitions and historical usage of "soluble" and "solvable" in mathematics, providing context for their interchangeability.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the terminology or the implications of the terms "soluble" and "solvable." Multiple viewpoints regarding language differences and historical context remain present.

Contextual Notes

Some statements rely on historical references that may not be universally accepted or verified, and the discussion includes assumptions about language use that are not fully explored.

pivoxa15
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How does soluble groups fit into algebra? Why is there another name for it called solvable groups?

What branch does it fall under?
 
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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.
 

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