What is the difference between the order of a group and the order of the elements of the group

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: "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." –  Anthony Labarre Mar 15 '11 at 6:52
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. –  Anthony Labarre Mar 15 '11 at 6:54
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. –  yunone Mar 15 '11 at 6:55
yunone, thankyou. I was thinking of multiplicative inverses. What are the "usual inverses" in $\mathbb{Z}$? –  Tyler Hilton Mar 15 '11 at 7:01
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. –  yunone Mar 15 '11 at 7:32

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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