In I.N Herstein's Topics in Algebra is given certain corollaries which follow from a definition.
Quoting:
COROLLARY 2 $\ \ \ \ \ \ \ \ \ $If G is a finite group and $a \in G$, then $a^{o(G)}=e$
This is clear to me.
The book then goes on to talk about how for an integer $n$, numbers less than $n$ and relatively prime to $n$ form a group under multiplication mod $n$ with order $\phi(n)$
Firstly. I do not understand what is meant by "multiplication mod n" and
Secondly, how corollary 3 (given below) follows from corollary 2 using the above group.
COROLLARY 3$\ \ \ \ \ \ \ \ \ $If n is a positive integer and a is relatively prime to $n$, then $a^{\phi(n)}\equiv1\ mod \ n$
Now, the Left-hand side in corollary 3 is the same as that of corollary 2 since for the group in which $a$ belongs the order is $\phi(n)$ as mentioned above.
This then means that the RHS in corollary should be identity.
So, I think my issue lies with the meaning of "multiplication mod n" and how for this binary operation 1 mod n is identity.