← All courses

Groups and Rings

Year 2 · 7 sections · 69 concepts · 158 practice questions

Start free (beta)Try chapter 1 free, no sign-up →

What's covered

free1 · Groups revisited10 concepts

Overview 1 · The plan of this section
Definition 1.1 · Subgroup generated by a set
Theorem 1.2 · Classification of cyclic groups
Proposition 1.3 · Subgroups of cyclic groups
Proposition 1.4 · The dihedral group by generators and relations
Proposition 1.5 · Conjugacy in Sₙ is cycle type
Definition 1.6 · Direct products
Proposition 1.7 · When is a product of cyclic groups cyclic?
Proposition 1.8 · Cosets and the index
Check 1.9 · Check: identifying a small group

2 · Normal subgroups and quotients9 concepts

Overview 2 · The plan of this section
Definition 2.1 · Normal subgroups
Proposition 2.2 · Tests for normality
Proposition 2.3 · Kernels are normal
Theorem 2.4 · Quotient groups
Theorem 2.5 · First isomorphism theorem
Theorem 2.6 · Second isomorphism theorem
Theorem 2.7 · Third isomorphism and correspondence theorems
Check 2.8 · Check: identifying a quotient group

3 · Group actions10 concepts

Overview 3 · The plan of this section
Definition 3.1 · Group actions, orbits and stabilisers
Proposition 3.2 · Actions are homomorphisms to Sym(X)
Corollary 3.3 · Cayley's theorem
Proposition 3.4 · Orbits partition the set
Theorem 3.5 · Orbit–stabiliser theorem
Theorem 3.6 · The class equation
Corollary 3.7 · p-groups have nontrivial centre
Theorem 3.8 · Burnside's lemma
Check 3.9 · Check: using actions to count

4 · Sylow theory and structure9 concepts

Overview 4 · The plan of this section
Theorem 4.1 · Cauchy's theorem
Definition 4.2 · Sylow p-subgroups
Theorem 4.3 · The Sylow theorems
Corollary 4.4 · A unique Sylow subgroup is normal
Proposition 4.5 · Groups of order pq
Example 4.6 · No simple group of order 12 or 30
Theorem 4.7 · Structure of finite abelian groups
Check 4.8 · Check: Sylow arguments

5 · Rings and ideals10 concepts

Overview 5 · The plan of this section
Definition 5.1 · Rings, integral domains and fields
Example 5.2 · Standard rings
Definition 5.3 · Subrings and ideals
Definition 5.4 · Ring homomorphisms
Proposition 5.5 · Kernels are ideals
Theorem 5.6 · Quotient rings
Theorem 5.7 · First isomorphism theorem for rings
Proposition 5.8 · Characteristic
Check 5.9 · Check: working with quotient rings

6 · Integral domains and factorisation10 concepts

Overview 6 · The plan of this section
Proposition 6.1 · Field of fractions
Definition 6.2 · Irreducible and prime elements
Proposition 6.3 · Prime implies irreducible
Definition 6.4 · Euclidean domains and PIDs
Theorem 6.5 · Euclidean domains are PIDs
Proposition 6.6 · In a PID, irreducible implies prime
Theorem 6.7 · PIDs have unique factorisation
Theorem 6.8 · Gauss's lemma
Check 6.9 · Check: factorisation in a ring

7 · Polynomial rings and fields11 concepts

Overview 7 · The plan of this section
Definition 7.1 · Maximal and prime ideals
Theorem 7.2 · Quotients by maximal and prime ideals
Corollary 7.3 · F[x]/(f) is a field iff f is irreducible
Theorem 7.4 · Eisenstein's criterion
Proposition 7.5 · Reduction mod p and low-degree tests
Definition 7.6 · Field extensions and degree
Theorem 7.7 · The tower law
Proposition 7.8 · Minimal polynomials
Theorem 7.9 · Finite fields
Check 7.10 · Check: irreducibility and degrees

Try it: a free card from this module

This is exactly what studying looks like inside UniMaths.

Theorem 1.2 · Classification of cyclic groups

Cyclic ⇒ $\mathbb Z$ or $C_n$ . $|g^k| = n/\gcd(n, k)$ ; $\varphi(n)$ generators.

Proof

Learn all 69 concepts like this

Cards, step-by-step proofs, and 158 questions scheduled so you review right before you forget.

Start free (beta)Try chapter 1 first