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)