I cannot find a definition for the group of reduced residue classes. Are the multiplicative group of integers modulo n the same thing?
The integers coprime to n from the set ${\displaystyle \{0,1,\dots ,n-1\}}$ of $n$ non-negative integers form a group under multiplication modulo $n$, called the multiplicative group of integers modulo n. Is this the same as the reduced residue classes $n$.
Yes, the reduced residue class modulo $n$ are the remainders of $\{0,1,2,...,n-1\}$ that are relatively prime to $n$. This is equivalent to having a multiplicative inverse modulo $n$. The remainders $1$ and $n-1$ are always two of them.