Need an example to show the need to check the closure property for being a subgroup under binary operation $\star$ of $(G, \star)$.

Question

I am able to theoretically sink in the idea as given here, but still need few simple examples. I mean that examples be of real, integer, rational domain and binary operations defined on them; or of rotations. Need this to really be familiar with the intricacies involved with the idea.

Answer 1

A group is just a bag of elements with an operation defined between them. (And a couple of special properties)

A subgroup is just a group inside a group. For example the integers with addition are a group. Take your bag full of integers and pick some integers, putting them in a second bag. Is that second bag a subgroup of the integers? Only if the second bag itself is a group! But for the second bag to be a group, the binary operation must make sense inside that bag.

I.e. take the second bag to be $\{1, 3, 4\} $. Does integer addition make sense in the universe of the second bag? Not really, because $3+4 = 7 \not\in \{1,3,4\} $

That is why you need a subgroup to be closed under the binary operation.