Category:Group Theory

A

• [+] Abelian Groups‎ (2)

C

• [×] Characteristic Subgroups‎ (0)
• [×] Conjugacy‎ (0)
• [×] Cosets‎ (0)
• [+] Cyclic Groups‎ (1)

G

• [×] Group Actions‎ (0)
• [+] Group Direct Products‎ (1)

G cont.

• [+] Group Examples‎ (5)
• [+] Group Homomorphisms‎ (5)
• [×] Group Isomorphisms‎ (0)
• [×] Group Monomorphisms‎ (0)

L

• [×] Lie Groups‎ (0)

M

• [×] Modulo Arithmetic‎ (0)

N

• [+] Normal Series‎ (1)

N cont.

• [+] Normal Subgroups‎ (3)

P

• [+] P-Groups‎ (1)
• [+] Prime Groups‎ (1)

R

• [+] Ring Theory‎ (22)

S

• [×] Simple Groups‎ (0)
• [×] Solvable Groups‎ (0)
• [+] Subgroups‎ (1)

A

• Alternative Axioms for Groups

C

• Cancellable Finite Semigroup is Group
• Cancellation Laws
• Cauchy's Group Theorem
• Cayley's Theorem (Group Theory)
• Center is Intersection of Centralizers
• Center of Group of Prime Power Order is Non-Trivial
• Classification of Groups of Order up to 15
• Commutation Property in Group
• Commutativity of Powers in Group

D

• Division Laws for Groups
• Division Ring Non-Zero Elements form Group

E

• Element to the Power of Group Order
• Element to the Power of Multiple of Order
• Element to the Power of Remainder
• Epimorphism Preserves Groups
• Equal Powers of Finite Order Element
• Equivalent Statements for Congruence Modulo a Subgroup
• Even Order Group has Order 2 Element
• External Direct Product of Groups

F

• Finite Group Elements of Finite Order

G

• Generator of a Group
• Group Element Commutes with Inverse
• Group is Inverse Semigroup with Identity
• Group Not Empty
• Group Product Identity therefore Inverses
• Group with Zero Element is Trivial

H

• Homomorphism to Group Preserves Identity and Inverses

I

• Identities all Self-Inverse
• Identity of Group is Unique
• Identity Only Group Element Order 1
• Identity Only Idempotent Element in Group
• Induced Group
• Induced Ring
• Inverse Completion an Abelian Group
• Inverse Element is Power of Order Less 1
• Inverse Mapping in Induced Structure
• Inverse of an Inverse
• Inverse of Product
• Inverse Subset Group Product
• Inverses in Group are Unique
• Invertible Elements of Monoid form Group

L

• Lagrange's Theorem (Group Theory)
• Latin Square Property

N

• No Group has Two Order 2 Elements

O

• Odd Order Group Element is a Square
• Order of Conjugate
• Order of Element Divides Order of Finite Group
• Order of Element Equals Order of Inverse
• Order of Group Product
• Order of Subset Product with Singleton
• Ordered Group Equivalences

P

• Permutation on a Polynomial is a Group Action
• Pointwise Addition forms a Group

P cont.

• Power of Product of Commutative Elements in Group
• Power of Product with Inverse
• Powers of Commutative Elements in Groups
• Powers of Elements in Group Direct Product
• Powers of Group Elements
• Powers of Group Elements/Negative Index
• Powers of Group Elements/Product of Indices
• Powers of Group Elements/Sum of Indices
• Powers of Infinite Order Element
• Product with Inverse equals Identity iff Equality

R

• Regular Representations in Group are Permutations
• Results concerning Order of Element

S

• Schur's Theorem (Group Theory)
• Self-Inverse Elements that Commute
• Self-Inverse Order 2
• Set Equivalence of Regular Representations
• Set of Words Generates Group
• Socks-Shoes Property
• Definition:Soluble Group
• Subset Product is Subset of Generator
• Subset Product: Powerset Semigroup
• Sylow Subgroup is Hall Subgroup
• Sylow Theorems

W

• Word Metric is a Metric

Z

• Z-Module Associated with a Group