669:. 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.
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732:. 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.
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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.
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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.
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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
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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.)
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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.
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715:. 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.
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469:"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
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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.
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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
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Shouldn't an article talk about only one concept? I ask it because
Wikidata assumes that every item represents a specific concept.--
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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).
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as the symmetry number (symmetry order)? Or order of a symmetry group has a broader meaning than symmetry number?
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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).
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I have now removed the "split" hatnote from the article as I see little support for the suggested split.
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If a group has finite order, there are a finite number of elements in that group by definition.
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mean to ask whether a finite group was defined as a group all of whose elements are finite?
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These two concepts are closely related but they are two different concepts.--
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Ahhhhh. I see. I have forgotten how to distinguish capital and small letters.
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Isn't that a finite group is defined as a group whose order is finite? --
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Why does a any element of a finite group have a finite order?
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Furthermore, I have now added a hatnote to disambiguate
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Split into "Group order" and "Order of a group element"
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is abelian, that is, commutative. What is the problem?
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575:Stupid, stupid, stupid. :) Thank you. --
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