Smallest subgroup and $\langle a \rangle$. Gallian says in Chapter 3 of Contemporary Abstract Algebra that 

For any element $a$ of a group $G$, it is useful to think of $\langle a \rangle$ as the smallest subgroup of $G$ containing $a$.

But wouldn't this mean the set $\langle a \rangle$ is simply $\{ a, a^{-1}, e \}$?
 A: The set $\{a,a^{-1},e\}$ has an identity and is closed under inverses, but it isn't necessarily closed under the group's operation.
If $H$ is a subgroup and $a \in H$, you also need $a^2=aa\in H$ and $a^3=aaa \in H$ etc. Likewise, you need inverses for all of those elements (i.e. $a^{-1}, a^{-2}, \dots \in H$). 
So closure under the group's operation and inverses require at least $\langle a \rangle = \{ a^n \;|\; n \in \mathbb{Z} \} \subseteq H$. 
Once you know  $\langle a \rangle$ is itself a subgroup. This shows that it is the smallest subgroup containing $a$.
A: Groups are closed under their operations.  In other words if $a,b \in G$ then we know that $ab \in G$.
So if $a\in <a>$ then we know that $a*a=a^2 \in <a>$ and $a^2*a = a^3 \in <a>$ and inductively we know that $a^2 \in <a>$ for all $n \in \mathbb N$.  And as inverses must exist we know $a^{-n} \in G$.
So in general the group $<a> = \{.......a^{-3},a^{-2},a^{-1}, e,a,a^2,a^3,....\}$.  But the $a^i$ need not be distinct.  It could be possible for $a^i = a^k;i \ne j$.  An example is $\mathbb Z_6=\{0,1,2,3,4,5\}$ under modulo addition.  $2^5 = 2+2+2+2+2 = 4 = 2+2 = 2^2$.  So $\{......2^{-3}=0,2^{-2}=2, 2^{-1}=4, 0, 2, 2^2 = 4, 2^3 = 0,....\} = \{0,2,4\}$ is not infinite.
Now it is possible that $a^k =e$ for some number $k$.  Then $a^{-1} = a^{k-1}$  Example: In $\mathbb Z_6$, $2^3 = 2+2+2= 0$ and $2^{-1} = 4 = 2^{3-1}$.  In this case $<a> = \{0, a, a^2, a^3, .... a^{k-1}\}$. Example $<2> \subset \mathbb Z_{6} = \{0,2,2^2=2^{-1}=4\}$
But it's also possible that $a^k \ne e$ for any $k \ne 0$.  In this case $<a> = \{a^n|n \in \mathbb Z\}$.  Example: If the group is $\mathbb Z$ under addition.  Then $<7> = \{0, 7, -7, 14, -14, 21, -21, ....\} = \{7n| n \in \mathbb Z\}$.  Which is infinite.   
....
In general.  If $a^n= e$ and $a^k\ne e; 1\le k < n$ we so "the order of $a$" is $n$ and we write it as $|a| = n$.  If no such $n$ exists, we say $|a| = \infty$.
If $|a| = n$ then $<a> =\{0,a, a^2, ....., a^{n-1}\}$.
If $|a| = \infty$ then $<a> = \{a^k|k \in \mathbb Z\}$.
Notice $<a>$ always has $|a|$ units.
