Groups

Your one-sentence version of the idea. It builds on sets and functions.

Definition

A group is a set $G$ with an operation $\cdot :G\times G\to G$ such that:

1. Associativity: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$
2. Identity: there is $e\in G$ with $e\cdot a=a\cdot e=a$
3. Inverses: for each $a$ there is $a^{-1}$ with $a\cdot a^{-1}=a^{-1}\cdot a=e$

The intuition

Why are these the right axioms? What example made it click?

Try it: the integers mod 4

Is $\{0,1,2,3\}$ with addition mod 4 a group? What is the inverse of $3$?

Solution

Yes. Addition mod 4 is associative, $0$ is the identity, and every element has an inverse: $3+1=4\equiv 0$, so $3^{-1}=1$.

Where I got stuck

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