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I know the order of a group is the size of the group, ie the number of elements. But what does it mean for an element of that group to have order?

Also, what are the precise definitions for

1) "element of a finite order of a group" and 2) order of an element of a group (assuming that the element has finite order)

If I remember correctly, the order of $\mathbb{Z}$ is one, however the order of the elements in this group have order infinity. Why is that? (Also I dont think $\mathbb{Z}$ is a group in the first place, is it? )

I would also like to ask another question if thats okay:

What is a cyclic group (and a precise definition for it as well) ?

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    From [Wikipedia](http://en.wikipedia.org/wiki/Order_%28group_theory%29): "the order, sometimes period, of an element $a$ of a group is the smallest positive integer $m$ such that $a^m$ = $e$ (where $e$ denotes the identity element of the group, and $a^m$ denotes the product of $m$ copies of $a$). If no such $m$ exists, we say that a has infinite order. All elements of finite groups have finite order."2011-03-15
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    As to $\mathbb{Z}$ not being a group, this claim makes no sense: a group is a set **together with a specific operation** that must satisfy some properties.2011-03-15
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    To address your comment on $\mathbb{Z}$, $(\mathbb{Z},+)$ is indeed a group. It's closed, the identity is $0$, and each element has the usual inverse. It's not a field though, if that's what you're thinking, since multiplicative inverses don't exist.2011-03-15
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    yunone, thankyou. I was thinking of multiplicative inverses. What are the "usual inverses" in $\mathbb{Z}$?2011-03-15
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    for any $n\in\mathbb{Z}$, $-n$ is its usual inverse. So $-5$ is the inverse of $5$, $17$ is the inverse of $-17$, etc.2011-03-15

2 Answers 2

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An element $g \in G$ has order $n$ if $g^n = e$ ($n$ is the smallest positive integer for which this is true). Where $e$ is the identity. See this wikipedia articles for more detail.

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You can usually find out about this sort of thing on Wikipedia. In this case, the answers to your questions are here:

Order (group theory)

Cyclic group

Integers#Algebraic_properties