A new question
For years, algebra has meant finding a number. You isolate the variable, check the solution, and move on. Abstract algebra asks something different: forget any one answer — what are the rules of the game? Take a set of objects and a way to combine any two of them into a third. Study that combination on its own terms, and you discover that wildly different worlds — the integers under addition, the symmetries of a triangle, clock arithmetic — all share the same hidden skeleton.
The starting object is the binary operation. It takes two inputs from a set and returns one output — always inside the same set. Ordinary addition does this on the integers: feed it 3 and 5, it returns 8, which is still an integer. Multiplication does it too. So does “combine these two rotations of a square.” The word binary just means “two inputs”; the magic is in which rules the operation happens to obey.
Four rules an operation might keep
Once you have a closed operation, you interrogate it. Four properties keep coming up, and you already know them from arithmetic — abstract algebra just gives them sharper edges.
- Closure: for all a, b in the set, a ∗ b is also in the set. No escaping.
- [[associative-property|Associativity]]: (a ∗ b) ∗ c = a ∗ (b ∗ c). Grouping does not change the result, so you can drop parentheses.
- [[identity-element|Identity]]: there is an element e with a ∗ e = e ∗ a = a for every a. It leaves everything unchanged — 0 for addition, 1 for multiplication.
- Inverses: for each a there is an element that combines with it to give the identity — its undo.
A fifth property, [[commutative-property|commutativity]] (a ∗ b = b ∗ a), is optional and often fails: putting on socks then shoes is not the same as shoes then socks. When an operation keeps the first four rules we are one step from a group; whether it also commutes decides whether the group is *abelian*. Notice we never specified what a and b *are*. They could be numbers, motions, or matrices. That deliberate vagueness is the whole point — prove something from the rules alone and it holds in every world that obeys them.
Reading an operation table
When a set is small, you can write the entire operation as a table — every input pair, every output, all at once. Here is addition on the clock of remainders mod 4, written a ⊕ b = (a + b) mod 4. The cell in row a, column b holds a ⊕ b.
Addition mod 4 on the set {0, 1, 2, 3}
⊕ | 0 1 2 3
---+------------
0 | 0 1 2 3
1 | 1 2 3 0
2 | 2 3 0 1
3 | 3 0 1 2
Read it: 2 ⊕ 3 = (2+3) mod 4 = 5 mod 4 = 1.
Checks you can see at a glance:
closure — every entry is a 0,1,2,3 ✓
identity — row 0 and column 0 copy the headers, so 0 is the identity ✓
inverses — a 0 appears in every row, so each element has an inverse
0↔0, 1↔3, 2↔2, 3↔1 ✓
symmetric across the diagonal — the table is mirror-symmetric, so ⊕ commutes ✓