What does "order" mean in group theory? For example, if I have the question:
"Find the primary decomposition of the abelian group 
$$
\mathrm{Aut}(C_{6125}).
$$
Compute the number of elements of order 35 in this group."
I know how to answer this question, but I don't understand what I'm looking for. What exactly does order mean?
 A: Definition: Let $G$ be a group and let $g\in G$. Then the order of $g$ is the smallest natural number $n$ such that $g^n = e$ (the identity element in the group). (Note that this $n$ might not exist).
So in your group, you are looking for all the elements $g$ that satisfy that 


*

*$g^{35} = e$

*$g^m \neq e$ for all $m<35$.


As mention in the comments below this answer, also beware that there is another notion of order in group theory. If $G$ is a finite group, then the number of elements in the group is called the order of the group. (If a group has infinitely many elements, then the group is sometimes said to have infinite order).
A: The question is asking about the number of elements $x\in Aut(C_{6125})$ such that the least positive integer $n$ such that $x^n=e$  is $35$. 
A: Using multiplicative notation, for a finite group $G$, the order of an element $g\in G$ is $\text{ord}(g) = n$ iff $g^n = e, n \in \mathbb{Z}, n>1$ and such that for all $m\neq n$, if $g^m = e \rightarrow m\ge n$. 
For a finite additive group $G$, $\text{ord}(g\in G) = n$, where $n$ is the least positive integer such that $ng = 0$.
In general, given a group $G$, if there is no positive integer $n$ such that $g^n = e$  for a given $g\in G$ (additively, such that $ng = e$), then $g$ is of infinite order, and the converse is also true. 
The order of an element in a group $G$ can also be thought of as equal to the order of the subgroup it generates: for $g\in G, \text{ord}(g) = |\langle g \rangle|$.
