REFERENCE SHEET

Groups & subgroups

A group isolates the algebra of reversible operations.

Before this Proof techniques

|G| = [G:H] · |H| (G finite)

Definitions

Group

A set G with an associative binary operation, an identity e, and an inverse for every element.

Subgroup

A subset H⊆G that is itself a group under the same operation.

Order

The order of a finite group is its size. The order of g is the least positive n with gⁿ=e, if such n exists.

Key theorems

Subgroup test

A nonempty subset H of a group is a subgroup if ab⁻¹∈H for every a,b∈H.

Lagrange theorem

For a finite group and subgroup H, the left cosets partition G into equal-sized sets, giving |G|=[G:H]|H|. In particular, element orders divide |G|.

Prime-order groups

Every group of prime order is cyclic.

Notable applications

Symmetry

Rotations and reflections of a geometric object compose into a group.

Modular arithmetic

Invertible residue classes form a multiplicative group used in arithmetic algorithms.

Work it out

Try before revealing

01 / Compute

Find the order of 4 in the additive group ℤ/12ℤ.

Hint

Find the smallest n>0 with 4n≡0 mod 12.

Solution

The order is 3: the generated subgroup is {0,4,8}.

02 / Prove

Show a group G of prime order p is cyclic.

Hint

Take g≠e and use Lagrange.