This problem bothers me for about two weeks.
Let $\mathcal{C}$ be the collections of all finite groups. Is $\mathcal{C}$ a primitive recursive set ?
I think we can embed it to a set which is primitive recursive if we want ro show $\mathcal{C}$ is primitive recursive. But I don't know which set to be chosen.
Any suggestions or comments are welcomed !
The question in this form is ill-posed.
The problem is that recursive sets are subsets of natural numbers or strings whose characteristic function is primitive recursive.
In order to make meaningful your question you have to specify the way you encode the structure of group as a number or a string.
For instance if you encode a group as a binary string representing the (underlying relation of the) group operation then the set is primitive recursive: a program that computes if the given string does represent a group operation has to compute the size of the group (which is the squared root of the string's size) and then it has do some finite checks that can be done in a polynomial time with respect of the dimension of the group, hence also with respect to the dimension of the string encoding the operation, hence they can be done with for loops.
Hope this helps.