Algebra III

Syllabus

• Groups and symmetries: definition, identity and inverses; abelian groups; additive vs multiplicative notation; examples (vector spaces, rings, GL(n), plane isometries).
• Cayley tables; classical groups: Klein group, dihedral groups, symmetric and alternating groups; cycle decomposition of permutations.
• Subgroups: closure tests; intersections; generated subgroups and writing elements in generated subgroups (abelian and general case).
• Cyclic groups: classification, order of an element, subgroups of cyclic groups; basic order formulas.
• Cosets and index; Lagrange theorem and consequences (incl. Euler–Fermat); prime order groups are cyclic; index-2 subgroups are normal.
• Homomorphisms: kernel and image; conjugation and automorphisms; normal subgroups; quotient groups; isomorphism theorem; simple groups (statements).
• Direct products: element orders, when a direct product of cyclic groups is cyclic; structure theorem for finite abelian groups (statement and use).
• Group actions: orbits and stabilizers; orbit–stabilizer theorem; Burnside lemma and a counting application; Cayley’s theorem (statement).
• Rings: units, zero divisors, characteristic and prime field; ideals and quotient rings; homomorphism theorem for rings; PID examples.
• Fields and extensions (overview): K[x]/(f) is a field iff f is irreducible; algebraic elements and minimal polynomials; splitting fields and normal extensions (statements).
• Finite fields (overview): size is a prime power; existence/uniqueness up to isomorphism; basic structure of additive/multiplicative groups.
• Selected applications and enrichments (high-level): coding theory ideas; quaternions and rotations; classical matrix groups (unitary/orthogonal) as symmetry groups.