Groups

前置知识

13 项必需的前置知识（一直追溯到底）

你将学会

• 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.

定义

• 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.

定理

• 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.

常见误解

• 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.

通向

可选发现

• 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.
• Geometric group theory A finitely generated group can be studied as a geometric object through its Cayley graph.
• ← 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

学习路径

• Undergraduate mathematics major
• Graduate preparation
• Geometry and topology
• Algebra and number theory
• Mathematical physics

延伸阅读

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