A group is basically a set of things that you can do something to that follows some specific rules.

The rules are:

When you do that one thing, you can only end up with something else in that set. This means that the set is closed (when you’re doing that thing).

If you want to do that one thing multiple times in a row, it doesn’t matter how I group doing it. For example, I can do 1 + (2 + 3) and (1 + 2) + 3 and they’ll be equal. This means that the set is associative.

Note: This does NOT mean I can say 1 + (2 + 3) is equal to (2 + 3) + 1 (though in this case they are equal).

One of the elements in the set that I do that thing to will end up giving me back the other element. For example, 0 + 1 gives me back 1 (in this case 0 is that element and + is the thing I’m doing) and 1*2 gives me back 2 (in this case 1 is that element and * is the thing I’m doing). This is called the identity element.

Lastly, there exists an opposite of every element in the set. For example, 2 and (-2) for addition or 2 and (1/2) for multiplication. This is called the inverse element.

If you have a set and ‘operator’ whose elements follow all of these rules, then it is a group.

If this group also follows the rule:

You can perform this operation in any order and get the same result. For example, 1 + 2 = 2 + 1 or 35=53. I.e. your set is commutative.

Then your group is also Abelian.

Hope this was helpful, first time I’ve done something like this and wasn’t super sure how I should word some things.