The model includes an oracle that executes the group operation.
One of the main uses of the generic group model is to analyse computational hardness assumptions.
An analysis in the generic group model can answer the question: "What is the fastest generic algorithm for breaking a cryptographic hardness assumption".
This question was answered for the discrete logarithm problem by Victor Shoup using the generic group model.
In particular, it has been shown[5] using a similar argument[6] that there exist cryptographic schemes which are provably secure in the generic group model but which are trivially insecure once the random group encoding is replaced with an efficiently computable instantiation of the encoding function.