Talk:Order (group theory)

Why does a any element of a finite group have a finite order?

Are there any WP articles currently which explain why any element of a finite group has finite order? (I don't need it explained to me; I'm just concerned it doesn't seem to be discussed anywhere, and is not immediately obvious to a group theory newcomer.) Dmharvey File:User dmharvey sig.png Talk 5 July 2005 18:14 (UTC)

Isn't that a finite group is defined as a group whose order is finite? -- Taku 11:00, 5 October 2005 (UTC)

No, a finite group is one which has a finite number of elements. It follows that every element has finite order, so it may mean the same thing, but it's not, strictly speaking, defined as such. J•A•K 13:19, 9 December 2005 (UTC)

Erm, the order of a group is the number of elements. Did Taku mean to ask whether a finite group was defined as a group all of whose elements are finite? Shawn M. O'Hare 23:20, 13 January 2006 (UTC)

Moving to finite group?

Am I the only one who feels strange about the fact that many well-known facts about finite groups (like how a group order is divided) are put here instead of finite group? Since when the order is infinite, little can be said so why don't we move the stuff to finite group and redirect this to group (mathematics)? -- Taku 11:19, 5 October 2005 (UTC)

Yes

Clarification, part 1

This might help clarify:

If a group has finite order, there are a finite number of elements in that group by definition.

For an element a in group G with identity e, consider the set {m is a natural number: a^m = e}. If this set is not empty, then by the Well-Ordering Principle, it has a least member. This least member is defined as the order of a.

Clarification, part 2

If G is a finite group of order n, and d is a divisor of n, then the number of elements in G of order d is a multiple of φ(d).

That should be:

If G is a finite group of order n, and d is a divisor of n, then the number of elements in G of order d is φ(d).

A Question

"If every element in G is the same as its inverse (i.e., g = g-1), then ord(g) = 2 and consequently G is abelian since ab = (bb)ab(aa) = b(ba)(ba)a = ba." Well...what about $Z_{2}\times Z_{2}$? --VKokielov 02:21, 8 July 2006 (UTC)

The quote is true and $Z_{2}\times Z_{2}$ is abelian, that is, commutative. What is the problem? JRSpriggs 02:54, 8 July 2006 (UTC)

Ahhhhh. I see. I have forgotten how to distinguish capital and small letters.

Stupid, stupid, stupid. :) Thank you. --VKokielov 03:28, 8 July 2006 (UTC)

A "capital letter" is also called an "upper-case letter", and a "small letter" is also called a "lower-case letter". I mention this because I have had college students in my courses, juniors and seniors, even, who did not know this.

Also, calling them "little letters" or "small letters", rather then "lower-case letters", has a childish sound to it. I call them "lower-case letters", personally. 98.81.17.64 (talk) 22:07, 4 August 2010 (UTC)

Order of symmetry group vs. symmetry number

Is the order of the symmetry group the same as the symmetry number (symmetry order)? Or order of a symmetry group has a broader meaning than symmetry number? Kazkaskazkasako (talk) 11:39, 2 June 2010 (UTC)

Split into "Group order" and "Order of a group element"

These two concepts are closely related but they are two different concepts.--Malore (talk) 16:17, 31 May 2018 (UTC)

• Oppose. The article is short. So all relevant properties should be easy to find if the article would be well written and well structured, which it is not. If you split the article, where will you put the fundamental result that the order of an element divides the order of the group? In any case, before splitting, the article should be completely rewritten for preparing the split. Therefore, the article must first be restructured and rewritten for having different sections for the properties of orders of groups and properties of orders of elements. When this will be done, and only then, one will be able to decide whether a split is useful. D.Lazard (talk) 16:47, 31 May 2018 (UTC)

• @D.Lazard: Shouldn't an article talk about only one concept? I ask it because Wikidata assumes that every item represents a specific concept.--Malore (talk) 00:57, 1 June 2018 (UTC)

• Oppose. I think the ideas are closely related enough, and that the resulting articles will be too short. Maybe if the article grows large but I don't see that happening any time soon. Wqwt (talk) 04:29, 19 September 2018 (UTC)
• Comment. The lede is very confusing. The lede says there are two meanings, but three things are discussed: The order of a group, the order of a group element, and the sequence number of an element in an ordered group. The lede should be restructured to make that clear. Comfr (talk) 04:17, 25 March 2019 (UTC)
• Oppose The two are closely related, most things that can be said about orders of elements can be generalized to orders of subgroups, while the concept of 'order of a group' alone is too trivial to merit an article. I've tried to clarify the distinction in the lead. Tokenzero (talk) 12:11, 14 May 2019 (UTC)

I have now removed the "split" hatnote from the article as I see little support for the suggested split. Tea2min (talk) 08:20, 3 January 2020 (UTC)

Furthermore, I have now added a hatnote to disambiguate Order (group theory) and Partially ordered group and Totally ordered group. Tea2min (talk) 08:35, 3 January 2020 (UTC)