Lecture · 16 min · advanced

Groups

Symmetry, made algebraic

An invitation to abstract algebra

The integers under addition, the nonzero reals under multiplication, and the symmetries of an equilateral triangle are the same kind of object: a set with an associative operation, an identity, and inverses. A group is that object. Once you see it, half of algebra is “which groups arise here?”

On the board today

—Check the group axioms on a familiar example.
—Distinguish abelian from non-abelian.
—Compute in a small symmetric group.

Axioms

associative \cdot, \exists e, \forall g \exists g^{- 1}, g \cdot e = e \cdot g = g

Closure is part of “the operation is a function $G \times G \to G$ .”

$(Z, +)$ is a group; $(N, +)$ is not (no inverses). $(R, \times)$ is not (zero has no inverse); $(R ∖ {0}, \times)$ is. The symmetries of the triangle, $S_{3}$ , is the smallest non-abelian group: a flip then a rotation is not a rotation then a flip. Abelian means $ab = ba$ for all pairs.

Worked example · Units modulo $5$

Show that ${1, 2, 3, 4}$ under multiplication modulo $5$ is a group.

1.Closed: products mod $5$ stay in the set.
2.Identity $1$ . Inverses: $2 \cdot 3 = 6 \equiv 1$ , $4 \cdot 4 \equiv 1$ .

This is $U (5)$ , cyclic of order $4$ , generated by $2$ .

Pause the lecture

Why is $({0, 1, 2, 3}, + mod 4)$ a group, while $({0, 1, 2, 3}, \times mod 4)$ is not?

Common mistake. Assuming every group is abelian because the first examples are. Always check $ab$ versus $ba$ . Matrix groups and symmetry groups are the usual counterexamples.

Practice this lecture