Yes, exactly. In group theory when we speak of a quotient group modulo one of its normal subgroups, we’re referring to exactly this. Z3, the integers mod 3, is isomorphic to (actually equal to) Z/3Z, the integers quotiented by all the multiples of 3.
People who haven’t studied abstract algebra aren’t used to thinking of it this way though. To most people, the modulus operator is specifically an operator that returns the remainder of Euclidean division. To think of it as a bunch of equivalence classes that split up the integers is not something most people think of right away.
Do you have a good resource for an intro to group theory and abstract algebra? I’ve never had a chance to study it in a course but it pops up a lot in physics and other interesting phenomena.
The best resource I know of is the textbook I bought for the course I took on group theory and ring theory [1]. It’s pretty expensive and the exercises are very challenging but if you’re a self-motivated student, you can learn a TON of abstract algebra from this one book. You may want to review some linear algebra before you dive in, if you haven’t done so in a while. You can find solutions to many of the exercises online though I can’t vouch for their accuracy.
People who haven’t studied abstract algebra aren’t used to thinking of it this way though. To most people, the modulus operator is specifically an operator that returns the remainder of Euclidean division. To think of it as a bunch of equivalence classes that split up the integers is not something most people think of right away.