I'm reading Rosen's book and it has a proof to show that a finite subgroup (set) is closed under a composition law.
It says for some $i$ and $j$, $i < j$, $a^i = a^j$ i.e, $a^i = a^i \circ a^{j-1}$. I really don't get it.
How is this equal? Is the pigeonhole principle applicable to all groups?
The pigeon hole principle says that for $n$ things grouped into $m$ distinct sets, at least one set will contain $\lceil n/m \rceil$ elements. Here, the elements are the distinct integer powers that we're raising $a$ to, and they're put in different sets according to what element of the group the $a^i$ corresponds to.
So, if there are more powers than elements of the group, eventually more than one power will be associated with the same element. So, $a^i = a^j = a^i a^{j-i}$. You have a $-1$ there, and that's erroneous.
Edit to add: Notice that this doesn't work for infinite groups, because you will never have more powers than elements of the group.
For example, in the group $(\Bbb Z, +)$, we will never have $n^i = n^j$ for $i \ne j$, because in this group the notation $n^i$ stands for $n \cdot i \in \Bbb Z$.