Definition talk:Order of Group Element
Doesn't this notion apply equally well to monoids? Relatedly, what do you call the equivalent to a torsion element in the realm of mappings? That is, what would you call a mapping $f$ whose composition with itself some number of times is the identity mapping? --Dfeuer (talk) 16:36, 14 February 2013 (UTC)
- Yes it does, but the context in which it is most usually first encountered is in the field of group theory. Students of abstract algebra usually study group theory first (maybe starting with semigroups), only learning what a "monoid" is later in their studies (if at all - many abstract algebra texts I have don't use the term at all).
- Raise another page defining the order of the element of a monoid by all means. --prime mover (talk) 17:57, 14 February 2013 (UTC)
- "... what would you call a mapping $f$ whose composition with itself some number of times is the identity mapping?" Since the algebraic structure consisting of mappings on a set forms a monoid, it's clearly an example of the above "order of monoid element". But such does not usually get raised as a separate issue because the concept in this context is not used so much. Except when it's in a ring, of course, then it's of paramount importance. --prime mover (talk) 18:00, 14 February 2013 (UTC)
- I only raised that question because Lord_Farin created Continuous Involution is Homeomorphism, and a slightly more general statement is that a continuous mapping of finite order (in this sense) is a homeomorphism. --Dfeuer (talk) 18:46, 14 February 2013 (UTC)
- Under the current terminology such mappings may be called "nilpotent", no? It's just awkward because the "zero" element isn't really the zero mapping. But abstract-algebraically that's the (or a) term. Also possible is "of finite order" as you mention. --Lord_Farin (talk) 21:41, 14 February 2013 (UTC)
- "I am no algebra expert" - it is expected that people contribute to this site according to the areas of mathematics which they know well, or at least have a reliable set of source works to back up their work. --prime mover (talk) 22:47, 14 February 2013 (UTC)
...at least until PW has become sufficiently comprehensive to serve as a reliable source work (or an amalgamation thereof) itself. At which point the same applies to the derived and deeper fields reached from basic algebra. Who knows - we might even get there some day. If only we posted more proofs... --Lord_Farin (talk) 22:52, 14 February 2013 (UTC)
Infinite order and the equivalence proof
Not all sources at Definition:Order of Group Element/Infinite use the definition there: Warner and Clark use def 2, Humphreys 1&2. They all define order and (in)finite order at once. Conclusion, I would put the definitions of infinite order with those of order and prove their equivalence at once. --barto (talk) (contribs) 12:57, 21 January 2018 (EST)
- Put a flag up, I'll revisit it. --prime mover (talk) 14:49, 21 January 2018 (EST)