Difference between revisions of "Group Theory (subject)"

From Maths

Latest revision as of 14:43, 26 February 2016

(Unknown grade)

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Here are some pointers of where to get started

v • d • e Group theory

Overview of the basic and important objects of group theory - a branch of abstract algebra

Primitives	semigroup [ilmath]\supset[/ilmath] monoid [ilmath]\supset[/ilmath] group [ilmath]\supset[/ilmath] Abelian group (AKA: commutative group), group action

Morphisms	Group homomorphism, Group isomorphism

Subobjects	subgroup, normal subgroup, cosets

Important theorems	First isomorphism theorem, second isomorphism theorem, third isomorphism theorem (overview of the group isomorphism theorems)

Important groups	Symmetric group (Permutation group), Alternating group, Dihedral group, Quaternion group, Klein-4 group

v • d • e Abstract algebra

Overview of the basic and important objects of abstract algebra

Single operation objects	semigroup [ilmath]\supset[/ilmath] monoid [ilmath]\supset[/ilmath] group

Commutative single operation objects	Abelian semigroup Abelian monoid [ilmath]\supset[/ilmath] Abelian group

Related objects of semigroups (thus all monoids and groups too)	Entire semigroup: Centre, Subsemigroup, Order Of subset: Centraliser, stability, Generator Of element: Orbit, Generator, Order

Related objects of monoids (thus all groups too)	Submonoid

Related objects of groups	Subgroup, Normal subgroup, Index

Important group theorems	3 isomorphism theorems, Butterfly lemma

Mappings between semigroups (thus between monoids and groups too)	Homomorphism, The set [ilmath]\text{Hom}(X,Y)[/ilmath] of homomorphisms between [ilmath]X[/ilmath] and [ilmath]Y[/ilmath], Other morphisms

Constructing new semigroups from old	Direct product, Direct sum

Group constructions	Quotient group

Objects related to mappings between groups	Image, Kernel

Hidden category: