Math 336 – Advanced Abstract Algebra
Review for Midterm Exam
The midterm exam will consist in 3 problems and 2 theoretical questions.
Review the theory for each section and homework problems.
Groups
Def. binary operation, identity, inverse, group;
Def. isometry, symmetries of a figure, dihedral groups.
Properties: uniqueness of identity and inverse, cancellation law.
Examples: Z, Zn, Dn, Mn(R), GL2(R), determinant, inverse
Presentations by generators and relations; ex. D(2n)=<r,s | rel.>
Finite groups; subgroups
Def. order of an element, order of a group, subgroups
Properties: |<a>|=|a|;
Tests for subgroups;
Def.Center, centralizer;
Cyclic groups
Def. cyclic group G=<a>;interpretation of cyclic groups: image of ax map.
Generators of cyclic groups
Properties: any cyclic group is isomorphic to Z or Zn.
Normal subgroups and Quotient Groups
- Def. subgroups, left/right cosets and properties (partition of the group), Lagrange Theorem
- Normal subgroups, quotient group, canonical projection G->G/N.
Homomorphisms
- Def. homomorphism, isomorphism, kernel, image
- 1st Isomorphism Theorem (G/ker(f)=im(f))
- Action of G on H: homomorphism G->Aut(H).
New groups from Old
- Sugroups, quotients, direct-product, semi-direct product
- Examples: plane isometries (rotations act on translations by conjugation), dihedral group.
Classification of finitely generated Abelian groups
- Commutator, abelianization, torsion of a group
- FT of Abelian groups;
- Other factorizations: p-groups and factorization.
Math 336 – Review for Final Exam
Final exam will consist in 3 problems and 2 theoretical questions.
Review the theory for each section and homework problems.
Abelian groups and codes
- Hamming code, distance, block code, parity check code
- Linear encoding, dual code
Sylow’s Theorems
- 1,2& 3rd Syllow Theorem
- Cauchy theorem
Permutations (Set automorphisms)
- Permutations of a set Aut(S), cycles, transpositions, composition;
- Th. Aut(S) is a group (with proof); symmetric group Sn;
- Order of a cycle, disjoint cycles commute;
- Factorization into cycles.
- Factorization into transpositions; length of product unique mod 2; even / odd permutations;
- Alternating group An
- Conjugacy classes: definition, characterization: correspond to shapes of permutations (partitions of n).
Galois Theory
- Fields, field extensions, degree [E:F] is multiplicative, adjoining a root
- Automorphisms of a field, automorphism of a field extension,
- Separable polynomials, irreducible polynomials
- Splitting fields and properties [E:F]<= deg(P)!
- Galois extensions, Galois groups Aut(E/F) for a normal extension
- Degree of a splitting field [E:F]=|Aut(E/F)|
- Galois correspondence: fields <-> subgroups
- Applications to higher order polynomial equations: Cardano’s method; deg >=5 not solvable;
- Equations solvable by radicals
- Applications to ruler and compass constructions: trisecting pi/3 is not possible.