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

In summary, soluble 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". solvable means able to be solved, whereas soluble means having a lot of water soluble substances in it.
  • #1
pivoxa15
2,255
1
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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  • #2
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?
 
  • #3
That is hard to swallow. I've only heard soluble made reference as a chemistry term.
 
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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.
 
  • #5
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.
 
  • #6
Do you agree with me it is just a difference in language?
 
  • #7
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)
 
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Especially since Galois was french!
 
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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.
 

What are soluble groups in algebra?

Soluble groups in algebra are groups that can be broken down into simpler groups through a series of normal subgroups. These subgroups are called "solvable" because they can be solved or simplified using specific algebraic techniques.

What is the difference between soluble groups and non-soluble groups?

The main difference between soluble and non-soluble groups lies in their ability to be broken down into simpler groups. Soluble groups can be solved using algebraic techniques, while non-soluble groups cannot be simplified in the same way.

What is the significance of studying soluble groups in algebra?

Soluble groups have important implications in algebraic structures and group theory. They can help us understand the structure and properties of groups, and can be used to solve complex algebraic problems.

What branch of mathematics do soluble groups fall under?

Soluble groups fall under the branch of mathematics known as group theory, which deals with the study of groups and their properties. Within group theory, soluble groups are a specific area of focus.

What are some real-world applications of studying soluble groups in algebra?

The study of soluble groups has many applications in various fields, including cryptography, coding theory, and physics. For example, the properties of soluble groups can be used in coding and decoding messages, and in understanding the symmetries of physical systems.

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