TITLE: Why don't we need to check for associativity for a subgroup?
QUESTION [9 upvotes]: The textbooks I read say that it's obvious that associativity follows in a subgroup, but it doesn't explain why (since it's not obvious to me). We still have to check for closure under the operation, the identity element, and the inverse, but why does associativity immediately follow for a subgroup just because the group itself has it?

REPLY [4 votes]: All mammals are animals.
Cats are mammals.
Therefore cats are animals.

Right? Well, that's the same argument as:

All triplets $(a, b, c)$ of elements in the group satisfy associativity.
All triplets $(a, b, c)$ of elements in the subgroup are in the group.
Therefore all triplets $(a, b, c)$ of elements in the subgroup satisfy associativity.