Recently, I talked to a native speaker about the proper usage of the word “kindly”. I frequently use phrases like “kindly let us know whether you agree with the suggested approach” in business let...
Let is used for assigning a mathematical value to a symbol. "Suppose N is finite" has meaning while "Let N be finite" doesn't make sense. On the other hand, both "Let n = 1" and "Suppose n = 1" are acceptable, though the former is preferred. I can see a situation when you might use the latter.
meaning - When do we use "suppose" and when "let"? - English Language ...
I think that "Let A be a set, let B be a group, and let C be a number." is the most formal phrasing. Since this is a mathematically formal usage, I think that would be preferred, but I don't think any of the phrases you presented are wrong, and "Let A be a set, B a group, and C a number." is shorter and more succinct. On a separate note, we have a sister site for Math you might consult.
grammaticality - "Let A be a set, [let] B [be] a group" - English ...
The author has taken the (correct) 'don't go there', which is a correctly formed second person imperative, and turned it into a verb, and the Let's is turned into an injunction for us to "don't go there".
grammar - "Let's not go there" or "'let's don't go there" - English ...
"Let me know if the problem persists" sounds to me like it means "if the problem persists, let me know". Option 2 sounds like you are putting words in the speaker's mouth.