المرحلة الجامعية المتقدمةAlgebraمُخطَّط له

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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يعتمد على

Algebraic structures: monoids, groups and rings Relations, equivalence relations and partitions

عدد المتطلبات السابقة الإلزامية (حتى الأساس): 13

ستتعلّم

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.

تعريف

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.

مبرهنة

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.

مفاهيم خاطئة شائعة

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.

يقود إلى

Number theory Quotient groups and quotient rings Homotopy and the fundamental group Enumerative combinatorics Representation theory of finite groups

اكتشافات اختيارية

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

مسارات التعلّم

قراءات إضافية

Michael Artin, Algebra (1991)
David S. Dummit and Richard M. Foote, Abstract Algebra