Normal Sub-Group Homework Help, Assignment Help, Online Tutor, Tutoring

Homework Help

Homework Help
View Details

Assignment Help

Assignment Help
View Details

Online Tutoring

Online Tutoring
View Details

Home » Math Homework Help » Algebra Homework Help » Normal Sub-Group

Normal Sub-Group

A sub-group N of G is said to be a normal sub-group of G if for every g G and n N, gng^-1 N. The normal subgroup N of G is denoted by N Δ G.

Now, gNg^-1 = {gng-1, n N), for some g G.

Alternatively, a subgroup N is said to be a normal sub-group of G if and only if gNg^-1 ⊂ N for every g G.

It is important to note that for a group G, the sub-groups of G and (e) are always normal. These subgroups are called trivial or improper normal subgroup.

Simple group: A group G ≠ (e) is said to be simple group if it has no proper normal sub-groups.

Theorem: N is a normal subgroup of G if and only if gNg^-1 = N for every g G.

Proof: Now it is given that gNg^-1 = N for every g G. Also gNg-1 = N

gNg-1 ⊂ N

N is a normal sub-group in G.

Conversely: N is a normal subgroup in G gNg^-1 ⊂ N

and g^-1Ng = g^-1N (g^-1)^-1 ⊂ N (1)

Multiplying by g on right and g-1 on left of (1), we get

gg^-1N (g^-1)^-1 g^-1 ⊂ gNg^-1 eNe⊂gNg^-1

Submit Your Query ???

Assignment Help

Calculus Homework Help Algebra Homework Help Abstract Algebra Help Discrete Math Help

Topics

Real Number Absolute Value Addition Of Matrices Square Matrix Adjoint Algebraic Structures Alternating Series Linear Equations Determinants Archimedean Real Numbers Binary Operation Binary Relation In A Set Bounded, Unbounded Sets Cauchy Root Test Caylay Hamiltion Theorem Circular Permutation Common Roots Complex Numbers Complex Number Conjugate Conjugate Of A Matrix Constant Sequences Convergence Of A Sequence Cosets Cubic, Biquadratic Equations De Moivre Theorem Real Number Denseness Order 3 Determinants Differences Of Matrices Direct Sum Of Vector Subspaces Eigen Vector Elementary Matrices Matrix Elem. Transformations Equal Matrices Equal Roots Two Permutations Equity Equivalent Matrices Trigonometry Function Expansion Field Function Algebra Fundamental Theorem Gaussian Integer Geometric Series Group Ideals Quantity Increasing Roots Infinite Series Convergent Integers Inverse Of Square Matrix Inverses Of Elementary Matrices Iota, Imaginary Numbers Left-Right Identity Sequence Limit Points Linear Combination Vectors Span Linear Dependence, Independence Linear Homogeneous Equations Two Subspaces Linear Sum Matric Polynomial Matrix Linear Equation Matrix Inverse Matrix Multiplication Matrix Scalar Multiplication Method Of Difference Minors And Co-factors Multiplication Modulo P Normal Sub-Group Normalizer Or Centalizer Orbit Of Permutation Peano Axioms Permutation Function Pigeon Hole Principle Matrices Integral Powers Mathematical Induction Principal Two Determinants Product Two Permutations Product Properties Of Modulus Rank Of A Matrix Rational Numbers Rational, Integral Polynomial Reciprocal Roots Relation Of Sets Rings Of A Set Row By Column Matrix Sequence Series Series Of Positive Terms Series Partial Sum Sequence Subrings Sum Of A Series Cosine Series Sum Sum Of Sine Series Symmetric/Skew Symm. Matrices Roots Symmetric Functions Symmetric Set Degree N Synthetic Division Transformation In General Transformations Of Equations Transpose Of A Matrix Matrix Transposed Conjugate Transposition Complex Numbers Representation Vector Space Vector Sub-spaces Whole Numbers