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It's pretty common to call a group, ring or module free when it has a 'basis', but unlike other mathematical definitions whose names can be easily related to the concept they describe (e.g. the spectrum of an operator), the name free hardly tells me anything intuitively about the object itself. So, the questions are:

What's the origin of this terminology?

Is there any hint on what the original motivation was?

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    $\begingroup$ You think that spectrum easily connects to the concept it describes?? I'd say it's a pretty colorful nomenclature (pun intended). $\endgroup$ – Nikolaj-K Dec 25 '14 at 2:11
  • $\begingroup$ Well, it clearly resembles the use given by the physicists. From Wikipedia: In Latin spectrum means "image" or "apparition", including the meaning "spectre". $\endgroup$ – hjhjhj57 Dec 25 '14 at 2:29
  • $\begingroup$ I'd argue that this is only true once you internalized the mathematical notion denoted by "image". E.g. the most common operator will be a matrix, a block of numbers, and its "action" is (if you got a field of characteristic zero) on infinite different vectors and there just by multiplication of these vectors by a number. I don't really see how this "clearly resembles the use given by the physicists". $\endgroup$ – Nikolaj-K Dec 25 '14 at 2:37
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"Free" usually means "free of relations", "independent". For example, a free group is the set of all words in a given alphabet $$a, a^{-1}, b, b^{-1}, c, c^{-1}\ldots .$$ The group operation is concatenation of words, with cancellation of $xx^{-1}$ and $x^{-1}x$. the neutral element is the empty word.

Every finitely generated group can be presented in a similar way but with certain "relations" that is several words which are equal to the neutral word by definition.

Then we have a homeomorphism from the free group to any other group generated by the same alphabet. This situation can be formalized in terms of category theory, and in any category a "free object" can be defined using the appropriate morphism diagram.

Reference: S. Lang, Algebra.

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    $\begingroup$ And is there some reason to think these are "origin" or "original motivation" for the term? $\endgroup$ – Gerald Edgar Dec 17 '14 at 19:40

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