Formal group

A formal group is a group of things that follows some special rules called axioms. These rules are like a secret code that the group members use to talk to each other and do cool things together.

Imagine you and your friends have a special club where you follow certain rules like always wearing a hat, clapping twice before you start talking, and saying a secret password to enter. That's like a formal group!

But instead of hats and clapping, a formal group is based on some special math rules. These rules help you tell if two things in the group are equal, and help you combine things in the group to get new things (like adding or multiplying numbers).

An example of a formal group is the group of integers modulo some number (or "mod" for short). This group has a special rule that if you add or subtract a number and then divide by another number (called the "modulus"), you always get a whole number between 0 and the modulus minus 1.

So if you're in the mod 5 group, you can add numbers like 2 + 3 = 0, because if you add 2 and 3 you get 5, which is the same as 0 when you divide by 5. Cool, right? And if you add 4 to 1 in the mod 5 group, you get 0 again!

Formal groups are like secret math clubs where you and your friends get to use cool rules to do math in a special way.