Talk:Generating set of a group

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Non-generators[edit]

Could we have an example of a non-trivial non-generator? AxelBoldt 18:31 Oct 6, 2002 (UTC)

I was going to create a new entry for Frattini subgroup (Fr(G) is the intersection of all maximal subgroups of G) which would include examples, as well as a proof that Fr(G) is the set of non-generators; this was intended as a "teaser" to get people to jump to that link. A simple example would be the cyclic C_p², where p is prime; then p is a non-generator. User:chas_zzz_brown 18:52 Oct 6 02 (UTC)

Removed intro statement[edit]

I removed this statement from the intro:

Whilst this may appear to produce excessively long strings of terms, in fact, these strings often can be reduced.

This is true, I'm sure, but quite out of place when we're not even done defining things yet. If someone could expand and reword, it could be turned into a paragraph appearing later in the article. Deco 18:56, 14 July 2005 (UTC)

Number of elements term[edit]

What is the technical term for the number of elements of a (minimal) generating set?

I'm not aware of any. Around the web I see things like "size of the minimal [or minimum] generating set". Deco 03:03, 28 July 2005 (UTC)

A group may have minimal generating sets of several different cardinalities (and some infinite groups have no minimal generating set at all), so "number of elements of a minimal generating set" is not well-defined. I'm guessing that what you really meant is the least cardinality for a generating set of the group; I've seen this referred to as the generating rank of the group (at least for abelian groups). --Zundark 20:23, 26 November 2005 (UTC)

Maybe you should look at the paper: Nonorientable embeddings of groups. European J. Combin. 9 (1988) 445-461. There the authors distinguish between irredundant generating set and minimal generating set. Since they work with Cayley graphs they give involutary generators different weight when defining the rank of the group. Tomo 21:33, 26 November 2005 (UTC)

Why "finitely many"?[edit]

I don't see why the first line stipulates finitely many elements. Does this not mean that an infinite group is not generated by a set of its elements (for instance, that the integers are not generated by 1)? I would be inclined to drop this stipulation; any comments? Mcboogie 18:49, 3 October 2007 (UTC)

I would be inclined to drop it too. It's correct (if correctly understood), but redundant (since infinite products aren't defined), and the fact that you've misunderstood it shows that it's misleading. --Zundark 19:05, 3 October 2007 (UTC)

Multiplicities of Esses[edit]

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)

This is now fixed. JackSchmidt (talk) 19:32, 11 December 2009 (UTC)

Lead paragraph unnecessarily complicated[edit]

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)

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)

But since x⁻¹ is a member of { 1, x, xx, xxx, ... } (by the definition of a group), adding it explicitly has no effect. Inverses are redundant in this context, no? David spector (talk) 01:57, 16 December 2009 (UTC)

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)

"Product" used as the name of the group operator[edit]

This article uses the word "product" when referring to the group operator. I don't know much mathematics, but isn't it true that the arithmetic integer product operator (with identity element 1) cannot form a group since the inverse of most elements would not be members of the group (since they wouldn't be integers)? Thus, the simple intuitive model group consisting of the integers under addition would fail because of calling the operator "product" instead of "addition", "sum", or just something abstract like "operator", "combine", or "apply". The naturalness and intuitiveness of these models mitigate against the use of "product" as the group operator. When we write "xx" instead of "x+x" (for brevity), I recommend that we think of the operator as "concatenate" instead of "add". David spector (talk) 02:52, 11 December 2009 (UTC)

The integers are not a very good model group, though they are reasonable as a model abelian group. The integers form a group under addition, but not under multiplication. Besides elements like 1/2 being outside the group, there is also the problem of 1/0 not being very well behaved at all. One could change the word product in the opening sentence, but this will just make the sentence longer, not shorter. JackSchmidt (talk) 19:32, 11 December 2009 (UTC)

Unless you have more rationale than this, I respectfully disagree. Integers with addition are an excellent model for groups, precisely because this group is neither a ring nor a field. I suspect that many schools introduce groups by using integers with addition as an example. For me, thinking of a group as "like" integer addition, a ring as "like" integers with addition and multiplication, and a field as "like" reals with addition, subtraction, multiplication, and division seems very natural. Also, the identity element of a group should not be written as 1, but 0, for the same reason (that it is incorrect in the most intuitive real-world model). 0 naturally concatenates with x (for example, in some computer programming languages, unlike 1) to yield x; 0+x = x in ordinary arithmetic. It seems clear to me that the word product is not appropriate for a group operator. David spector (talk) 02:20, 16 December 2009 (UTC)

To be clear, I am assuming you mean "my rationale for using the word product in this particular article is that it will make the opening sentence longer to use apply/combine/etc." and that you do not think this is a good rationale for using "product" in this article. Changing the word is fine by me, but I think it will make that sentence longer. I just didn't see how to satisfy your two requests at once.

As far as model groups go, there are basically two completely unrelated algebraic structures: abelian groups and groups. Now the definitions look pretty similar, and they are often first taught together, but basically the human pursuit of studying these two concepts are no more related than any other two areas of algebra. In fact the study of rings and the study of abelian groups are intimately related, while the study of groups is not particularly closely related to the study of rings.

Your reasons for thinking the integers are a good model group all strike me as very good reasons for it to be a model *abelian group* (surely it should be the first abelian group, followed by finite cyclics, the additive rationals, the Prüfer groups, and finally the group of rational numbers whose denominators are square-free). I was initially trained in abelian groups and modules (abelian groups being acted upon by a fixed ring), and I think it is a great way to get introduced to algebra. It's just very, very different than group theory.

Good models for a plain old group are the group of invertible 2x2 matrices, groups of rotations (say the icosahedral group or a dihedral group), or the symmetric group (say on 3 points). Each of these groups has a natural action, either on vectors, on a geometric shape, or on "points". Group theory is very much concerned with groups acting on other things (consider Galois theory or Lie groups or invariant theory, the classical origins of group theory). Abelian group theory is very much concerned with abelian groups being acted upon by other things. JackSchmidt (talk) 07:17, 16 December 2009 (UTC)

I changed "product" in the opening paragraph. Certainly product is correct to a mathematician, but certainly calling it a product and then having so many additive examples will confuse readers. I chose "combination" just because it allowed for a relatively short and minor change that sounded fine. I thought about "combination" without qualification, but since it has some unrelated technical meanings (like choosing 3 generators out of the set of 4 without regards to order), I thought it was good to be explicit. Generally people like to downplay the operation and any sentence I came up with using "apply" or "operate" had to promote the operation to more than a parenthetical prepositional phrase. JackSchmidt (talk) 07:28, 16 December 2009 (UTC)

Relation to independence[edit]

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)

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)

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