Universidad avanzadaAlgebraPlanificado
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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Se basa en
Requisitos previos obligatorios, hasta el fondo: 13
Aprenderás a
- 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.
Definición
- 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.
Teorema
- 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.
Errores frecuentes
- 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.
Conduce a
Descubrimientos opcionales
- 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
Itinerarios de aprendizaje
Lecturas complementarias
- Michael Artin, Algebra (1991)
- David S. Dummit and Richard M. Foote, Abstract Algebra