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 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.
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 ...
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 ...
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.