Talk:Generating set of a group

message: Opening para's:
'In abstract algebra, a generating set of a group G is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses.
More generally, if S is a subset of a group G, then <S>, the subgroup generated by S, is the smallest subgroup of G containing every element of S, meaning the intersection over all subgroups containing the elements of S; equivalently, <S> is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses.'
Whilst possibly strictly true, the reuse of S for the more general case is very confusing; careful reading resolves the issue, but using any other letter would avoid the problem entirely. Alteratively why not avoid S in the opening para and say simply "...a generating set of a group is a subset such that every element of the group can be expressed as the product.... of elements of that subset and their inverses"
Julian I Do Stuff (talk) 13:17, 8 October 2009 (UTC)
replies: text: This is now fixed. JackSchmidt (talk) 19:32, 11 December 2009 (UTC)
thread_title:  Multiplicities of Esses 

message: The lead paragraph is unnecessarily complicated. While I cannot provide a shorter paragraph, since I am not a mathematician, I believe that, for example, the phrase "and their inverses" can be omitted, since it is unnecessary in the definition of a generating set. One-sentence definition of a generating set David spector (talk) 02:20, 11 December 2009 (UTC)
replies: text: Omitting "and their inverses" leads to several confusions.  For instance, an infinite cyclic group generated by {x} is not the set of finite products of elements of {x}.  That set is just { x, xx, xxx, ... } or { 1, x, xx, xxx, ... } if one allows empty products.  One also needs to use x−1. JackSchmidt (talk) 19:32, 11 December 2009 (UTC)

text: I'm afraid this is incorrect.  x−1 is not a member of { 1, x, xx, xxx, ... }.  { 1, x, xx, xxx, ... } is not a group.  In additive terms, I am saying if you start with 1 and add it to itself a bunch of times you get { 1, 2, 3, ... } or at best { 0, 1, 2, 3, ... } and -1 is not in { 0, 1, 2, 3, ... } and { 0, 1, 2, 3, ... } is not a group.  You want to explicitly include an additive inverse like -1 so that you can cancel the additions.  In a finite group this does not matter, but in infinite groups it can be a substantial problem.  JackSchmidt (talk) 06:25, 16 December 2009 (UTC)
thread_title:  Lead paragraph unnecessarily complicated 

message: This concept of a set of generators seems to me to be like the Group theory analogue of an algebraically independent set and a basis, or linearly independent set. Has anyone else noted this similarity or seen it noted? Further discussion would be interesting. LokiClock (talk) 14:32, 22 December 2009 (UTC)
replies: text: It's actually the analogue of a spanning set (there's no requirement for independence). Every type of algebraic structure has an analogous concept (usually called a generating set - the term "spanning set" for the vector space case is anomalous). --Zundark (talk) 15:04, 22 December 2009 (UTC)
thread_title:  Relation to independence