gh-40527: Refactor sage.groups.generic
    
Previously, the code to check `in addition_names`, `in
multiplication_names`, error checking etc. are duplicated everywhere.
Now it's collated into a helper function.

Besides, previously sometimes these group definition arguments are not
correctly passed to subclasses. Now they are.

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### ⌛ Dependencies

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URL: #40527
Reported by: user202729
Reviewer(s): Lorenz Panny, user202729

Author: Release Manager