Advanced undergraduateAlgebraPlanned

Groups

Groups and subgroups, cyclic, dihedral and symmetric groups, cosets and Lagrange's theorem, homomorphisms and normal subgroups, and group actions with the orbit–stabiliser theorem.

math.algebra.groups

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The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.

Builds on

Algebraic structures: monoids, groups and rings Relations, equivalence relations and partitions

Required prerequisites, all the way down: 13

You will learn to

Work with cyclic, dihedral and symmetric groups, computing orders of elements and subgroups.
Prove Lagrange's theorem using cosets as equivalence classes.
Identify kernels and images of homomorphisms and recognise normal subgroups.
Apply the orbit–stabiliser theorem and Burnside's lemma to counting problems.

Definition

A subgroup is a subset that contains the identity and is closed under the operation and under inverses.
The left coset of H containing g is gH = {gh : h ∈ H}.
A subgroup N is normal if gNg⁻¹ = N for every g in the group.
An action of G on a set X is a map G × X → X with e · x = x and g · (h · x) = (gh) · x.

Theorem

Lagrange: the order of a subgroup of a finite group divides the order of the group.
Every subgroup of a cyclic group is cyclic; up to isomorphism the cyclic groups are ℤ and ℤ/nℤ.
The kernels of group homomorphisms are exactly the normal subgroups.
Orbit–stabiliser: for a finite group, |G · x| = |G| / |G_x|.
Cayley: every group is isomorphic to a group of permutations of some set.

Common misconceptions

Believing the converse of Lagrange's theorem: A₄ has order 12 but no subgroup of order 6.
Thinking every subgroup is normal.
Assuming groups are commutative.

Leads to

Number theory Quotient groups and quotient rings Homotopy and the fundamental group Enumerative combinatorics Representation theory of finite groups

Optional discoveries

Functors An action of G on a set X is the same thing as a functor from the one-object category BG to the category of sets.
Cryptography Diffie–Hellman and elliptic-curve cryptography work in finite abelian groups in which discrete logarithms are believed to be hard to compute.
← Doing and undoing
← Symmetry: slides, turns and flips
← One thing after another
← Integers
← Clock arithmetic: remainders and modular arithmetic
← Structure-preserving maps: homomorphisms and isomorphisms
← Congruences
← Transformation geometry and the Erlangen programme