Abstract Algebra I
The theory of symmetry: the symmetries of polygons and the origins of groups in the study of polynomial equations, the group axioms and their first consequences, modular arithmetic, the groups of units and cyclic groups with their generators and subgroups, permutations in cycle notation, parity and the alternating group, dihedral groups by generators and relations and Leonardo's theorem on plane symmetry, subgroup tests, generated subgroups and subgroup lattices, cosets and Lagrange's theorem with Fermat's and Euler's theorems and the classification of groups of orders 4 and 6, homomorphisms, kernels, isomorphism invariants and Cayley's theorem, normal subgroups, quotient groups, direct products and the simplicity of the alternating group A5, the three isomorphism theorems, commutator subgroups and composition series, group actions with orbits, stabilizers, the class equation, Cauchy's theorem and Sylow's theorems, and a capstone on counting necklaces and painted cubes with Burnside's lemma and on the group of the Rubik's cube, with runnable Python examples throughout.
MATH-322 Year 3 3 credits 12 lessons · 4 h 37 min ▶ Video lessons coming soon
Offered by HSM University · Mathematics & Statistics
What you'll learn
- Describe the symmetries of a figure as a set closed under composition, compute products of symmetries of the triangle and the square, and explain how the idea of a group grew out of the study of polynomial equations.
- State the group axioms, verify them for number systems, matrices and permutations, and prove basic consequences: uniqueness of the identity and inverses, cancellation, the socks-and-shoes rule and the Latin square property of Cayley tables.
- Compute in the groups \(\mathbb{Z}_n\) and \(U(n)\), find element orders, prove that subgroups of cyclic groups are cyclic, and count the generators and subgroups of a cyclic group.
- Write permutations in cycle notation, compose and invert them, find their orders as least common multiples of cycle lengths, determine their parity, and work with the alternating group.
- Describe the dihedral group by generators and relations, calculate with rotations and reflections, find its center and conjugacy classes, and apply Leonardo’s theorem to plane figures.
- Use the one-step and finite subgroup tests, find centralizers, generated subgroups and product sets, and draw the subgroup lattices of small groups such as \(\mathbb{Z}_{12}\) and \(D_4\).
- Compute left and right cosets, prove Lagrange’s theorem and its corollaries, derive Fermat’s and Euler’s theorems from it, and classify the groups of orders 4 and 6.
- Decide whether a map is a homomorphism, find its kernel and image, prove or disprove that two groups are isomorphic using invariants, and prove Cayley’s theorem.
- Test subgroups for normality, construct quotient groups and prove the coset product is well defined, recognize internal direct products, and prove that \(A_5\) is simple.
- State and prove the first, second and third isomorphism theorems, use them to identify quotient groups, and compute commutator subgroups and composition series.
- Define group actions, compute orbits and stabilizers, apply the orbit–stabilizer theorem and the class equation, and prove Cauchy’s theorem and that \(p\)-groups have nontrivial centers.
- Count colourings, necklaces and other patterns up to symmetry with Burnside’s lemma, and explain the order of the Rubik’s cube group in terms of homomorphisms and invariants.
Course content
12 lessons · 4 h 37 min total · final exam
- 1 Symmetry and the idea of a group 24 min · quiz
- 2 Groups: definitions and first examples 23 min · quiz
- 3 Modular arithmetic groups and cyclic groups 22 min · quiz
- 4 Permutation groups and the symmetric group 21 min · quiz
- 5 Dihedral groups and symmetries of figures 23 min · quiz
- 6 Subgroups and generators 23 min · quiz
- 7 Cosets and Lagrange's theorem 24 min · quiz
- 8 Homomorphisms and isomorphisms 25 min · quiz
- 9 Normal subgroups and quotient groups 23 min · quiz
- 10 The isomorphism theorems 24 min · quiz
- 11 Group actions, orbits and stabilizers 23 min · quiz
- 12 Capstone: counting with symmetry and Burnside's lemma 22 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
The theory of symmetry: the symmetries of polygons and the origins of groups in the study of polynomial equations, the group axioms and their first consequences, modular arithmetic, the groups of units and cyclic groups with their generators and subgroups, permutations in cycle notation, parity and the alternating group, dihedral groups by generators and relations and Leonardo's theorem on plane symmetry, subgroup tests, generated subgroups and subgroup lattices, cosets and Lagrange's theorem with Fermat's and Euler's theorems and the classification of groups of orders 4 and 6, homomorphisms, kernels, isomorphism invariants and Cayley's theorem, normal subgroups, quotient groups, direct products and the simplicity of the alternating group A5, the three isomorphism theorems, commutator subgroups and composition series, group actions with orbits, stabilizers, the class equation, Cauchy's theorem and Sylow's theorems, and a capstone on counting necklaces and painted cubes with Burnside's lemma and on the group of the Rubik's cube, with runnable Python examples throughout.
Part of the Science stream, typically taken in year 3.