Identify independent clauses

If both sides of a sentence could each stand alone as a complete sentence, you have two independent clauses, and they need a strong enough join. There are three correct options: a period, a semicolon, or a comma followed by a coordinating conjunction.

The coordinating conjunctions are the FANBOYS words — for, and, nor, but, or, yet, so — and they must come right after the comma to hold two independent clauses together.

A comma alone is not strong enough; using one between two independent clauses is a comma splice. "Silent movies were popular, they relied on visual storytelling" is a comma splice, fixed with a semicolon, a period, or a comma plus "and."

So the check is simple: test each side of the punctuation. If both sides could stand alone, you need a period, a semicolon, or a comma plus a FANBOYS word. If one side is dependent, a comma may be enough.

How should these two independent clauses be joined: "The rivalry began early" and "the two teams have met many times since"?
Since both stand alone, use a period, a semicolon, or a comma plus a conjunction — for example, "The rivalry began early, and the two teams have met many times since."

Is "La Guardia was elected mayor in 1934, he mobilized the city during the war" correct?
No. Both parts are independent clauses joined by only a comma, which is a comma splice. Use a semicolon, or add "and" after the comma.

Two independent clauses: "A zombie star is not truly gone" and "it returns to life." Give a correct one-sentence join.
Use a semicolon: "A zombie star is not truly gone; it returns to life." Both sides stand alone, so a semicolon is appropriate.