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Group theory is all about symmetries. Can this be seen from the axioms defining a group? Or equivalently can the group axioms be motivated from this point of view? Of course one can look at several examples and check that the group axioms are fulfilled, nevertheless this doesn't make clear why the axioms have to be precisly like they are. Any ideas would be much appreciated.

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    $\begingroup$ "Doing nothing is a symmetry of an object" "Doing something and then doing that something in reverse is the same as doing nothing, and we can always do a something in reverse" seem pretty natural to me. $\endgroup$ – Dan Rust Jun 28 '14 at 9:24
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    $\begingroup$ Have a read of this essay: pauli.uni-muenster.de/~munsteg/arnold.html (motivational quote: "What is a group? Algebraists teach that this is supposedly a set with two operations that satisfy a load of easily-forgettable axioms. This definition provokes a natural protest: why would any sensible person need such pairs of operations?"), then have a look in the opening sections of "Abel's Theorem in Problems and Solutions" books.google.ie/books?id=GI_SmiYsh0UC&source=gbs_navlinks_s for the details. $\endgroup$ – bolbteppa Jun 28 '14 at 11:31
  • $\begingroup$ The essay is hilarious, thanks a lot! I will have a look at the book $\endgroup$ – jak Jun 28 '14 at 12:02
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The most general interpretation of symmetry is a bijection of a set onto itself. The group axioms model sets of such bijections under composition. Cayley's theorem shows this interpretation can be made concrete.

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  • $\begingroup$ Arnold's essay mentioned in the comments makes exactly the same point, in a somewhat different bias. $\endgroup$ – lhf Jun 28 '14 at 13:42

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