Abstract Algebra Definitions

Question	Answer
Order of permutation	the order of a permutation is the l.c.m. of the lengths of the cycles in its cycle decomposition.
Binary operation Associative Commutative	Image: Binary Operation, Associative, Commutative (binary/octet-stream)
Stabilizer of s in G	if G is a group acting on a set S and s is some fixed element of S, the stabilizer of s in G is the set Image: Stabalizer Of S In G (binary/octet-stream)
Group	A group is an order pair (G, ) where G is a set and is a binary operation on G satisfying the following axioms: Image: Group (binary/octet-stream)
Zero divisor & Unit in R	Image: Zero Divisor. Unit (binary/octet-stream)
Sylow's Theorem (formal)	Image: Sylow's Theorem(Formal) (binary/octet-stream)
Transposition (cycles)	Image: Transposition (Cycles) (binary/octet-stream)
Symmetric group on the set Omega Permutations of Omega	Image: Symmetric Group On The Set Omega, Permutations (binary/octet-stream)
Third Isomorphism Theorem (groups)	Image: Third Isomorphism Theorem (binary/octet-stream)
Unique Factorization Domain (U.F.D)	Image: Unique Factorization Domain (binary/octet-stream)
Subgroup of G generated by A	Image: Subgroup Of G Generated By A (binary/octet-stream)
Subring	Image: Subring (binary/octet-stream)
Subgroup	Image: Subgroup (binary/octet-stream)
Sylow Theorem	Image: Sylow Theorem (Groups) (binary/octet-stream)
Sum, Product, Power of Ideals I	Image: Sum, Product, Power Of Ideals (binary/octet-stream)
Solvable groups	Image: Solvable (Groups) (binary/octet-stream)
Sign of sigma Even and Odd Permutations	Image: Sing Of Sigma. Even And Odd Permutations (binary/octet-stream)
Simple groups	Image: Simple (Groups) (binary/octet-stream)
Second isomorphism theorem (groups)	Image: Second Isomorphism Theorem (binary/octet-stream)
Relations	Image: Relations (binary/octet-stream)
Quotient group or Factor group	Image: Quotient Group Of Factor Group (binary/octet-stream)
Quotient Ring	Image: Quotient Ring (binary/octet-stream)
Ring	Image: Ring (binary/octet-stream)
Second Isomorphism Theorem for Rings	Image: Second Isomorphism Theorem For Rings (binary/octet-stream)
Presentation of G	Image: Presentatino Of G (binary/octet-stream)
Principal Ideal Domain (P.I.D)	Image: Principal Ideal Domain (binary/octet-stream)
Prime Ideal	Image: Prime Ideal (binary/octet-stream)
Quaternion Group Q_8	Image: Quaternion Group Q8 (binary/octet-stream)
P-groups & Sylow p-subgroups	Image: P Groups. Sylow P Subgroups Of G. (binary/octet-stream)
Orbit of G & Transitive action	Image: Orbit Of G. Transitive (Action) (binary/octet-stream)
Order (element)	Image: Order (binary/octet-stream)
Norm & Positive Norm	Image: Norm, Positive Norm (binary/octet-stream)
Natural Projection (homomorphism) of G onto G/N	Image: Natural Projection (Homomorphism) Of G Onto Gn (binary/octet-stream)
Normalizer of A in G	Image: Normalizer Of A In G (binary/octet-stream)
Length of a Cylce	Image: Length Of A Cycle (binary/octet-stream)
Lattice of Subgroups of Q_8	Image: Lattice Of Subgroups Q 8 (binary/octet-stream)
Left and Right Cosets	Image: Left And Right Cosets (binary/octet-stream)
Maximal Ideal	Image: Maximal Ideal (binary/octet-stream)
Left, Right, Two-sided Ideal	Image: Left, Right, Two Sided Ideals (binary/octet-stream)
Lattice of D_8	Image: Lattice Of D 8 (binary/octet-stream)
Lagrange's Theorem	Image: Lagrange's Theorem (binary/octet-stream)
Lattice of S_3	Image: Lattice Of S 3 (binary/octet-stream)
Klein 4 Group	Image: Klein 4 Group (binary/octet-stream)
Lattice of D_16	Image: L Attice Of D 16 (binary/octet-stream)
Integral Domain	Image: Integral Domain (binary/octet-stream)
Kernel of a Homomorphism \rho	Image: Kernel Of A Homomorphism (binary/octet-stream)
Kernel of an Action	Image: Kernel Of An Action (binary/octet-stream)
Isomorphism	Image: Isomorphisms (Groups) (binary/octet-stream)
Irreducible, Prime, and Associate element	Image: Irreducible, Prime, Associate Elements (binary/octet-stream)
Homomorphism (groups)	Image: Homomorphism (Groups) (binary/octet-stream)
Inner automorphism	Image: Inner Automorphism (binary/octet-stream)
Index of H in G	Image: Index Of H In G (binary/octet-stream)
Ideal generated by A	Image: Ideal Generated By A. Principal Ideal. Finitely Generated Ideal (binary/octet-stream)
Field	Image: Field (binary/octet-stream)
First Isomorphism Theorem (groups)	Image: First Isomorphism Theorem (binary/octet-stream)
Group Action	Image: Group Action (binary/octet-stream)
First Isomorphism Theorem for Rings	Image: First Isomorphism Theorem For Rings (binary/octet-stream)
Fourth Isomorphism Theorem (Groups)	Image: Fourth Isomorphism Theorem (binary/octet-stream)
Euclidean Domain	Image: Euclidean Domain (binary/octet-stream)
Division Ring	Image: Division Ring (binary/octet-stream)
Fibers of \rho	Image: Fibers Of Rho (binary/octet-stream)
Cycle type of \sigma & Partition of n	Image: Cylce Type Of Sigma. Partition Of N (binary/octet-stream)
cyclic group	Image: Cyclic Group (binary/octet-stream)
Cycle decomposition of \sigma	Image: Cycle Decomposition Of Sigma (binary/octet-stream)
Conjugate (element , set) & Normal group	Image: Conjugate (Element And Set) (binary/octet-stream)
Conjugate (subsets)	Image: Conjugate (Subsets) (binary/octet-stream)
Conjugate (elements)	Image: Conjugate (Elements) (binary/octet-stream)
Class Equation	Image: Class Equation (binary/octet-stream)
Binary operation	Image: Binary Operation, Associative, Commutative (binary/octet-stream)
Cauchy's Theorem	Image: Cauchy's Theorem (binary/octet-stream)
Cayley's Theorem	Image: Cayley's Theorem (binary/octet-stream)
Cycle	Image: Cycle (binary/octet-stream)
Composition Series	Image: Composition Series (Sequence Of Subgroups) (binary/octet-stream)
Center of G	Image: Center Of G (binary/octet-stream)
Characteristic in G	Image: Characteristic In G (binary/octet-stream)
Centralizer of A in G	Image: Centralizer (binary/octet-stream)
Alternating group	Image: Alternating Group Of Deg N (binary/octet-stream)
Automorphism of G	Image: Automorphism Of G (binary/octet-stream)
Alternating Group Example	Image: Alternating Group Example (binary/octet-stream)

Order of permutation

the order of a permutation is the l.c.m. of the lengths of the cycles in its cycle decomposition.

Binary operation Associative Commutative

Image: Binary Operation, Associative, Commutative (binary/octet-stream)

Stabilizer of s in G

if G is a group acting on a set S and s is some fixed element of S, the stabilizer of s in G is the set

Image: Stabalizer Of S In G (binary/octet-stream)

Group

A group is an order pair (G, *) where G is a set and * is a binary operation on G satisfying the following axioms:

Image: Group (binary/octet-stream)

Zero divisor & Unit in R

Image: Zero Divisor. Unit (binary/octet-stream)