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Normal Subgroup {Algebra} (B.Sc. Mathematics NEP Notes)

 NNormal Subgroup


Normal Subgroup

    Let G be a group, and H be a subgroup of G. Then subgroup H is set to be normal subgroup of G if for any element g in & h in H s.t.

 ghg−1 Є H

 

Simple Group

    A simple group is a type of group in abstract algebra that has no non-trivial normal subgroups. In other words, a group G is simple if the only normal subgroups of G are the trivial subgroup {e} and the whole group G itself.

 

Hamiltonian Group

A non-abelian group in which each sub-Group is normal.


Conjugate Element

    In group theory, given a group G and elements a, b in G the element b is said to be conjugate to "a" if there exists an element g in G such that, b=gag −1

The process of finding such a g is called conjugation.


Normalizer of an element

    The normalizer of an element a in a group G, denoted as N(a), is defined as the set of all elements in G that, which commutes with a. The normalizer is given by:

N​(a)={g∈G ∣ ga=ag}

 

Centralizer

Let A be a non-empty subset of a group G. The centralizer C(A) of A, in G is defined as:

C(A) = {xєG: ax=xa ∀ a є A}


Conjugate Class

    In group theory, elements that are conjugate to each other form a conjugacy class. Given a group G and an element a in G, the conjugacy class of a, denoted as Cl(a), is the set of all elements in G that are conjugate to a.

Cl(a)={gag−1∣g∈G}.

 

Self-Conjugate

    An element a in a group is said to be self-conjugate if gag−1=a for all g in the group. In other words, the conjugacy class of a consists only of a itself. 

a is self-conjugate if {gag−1∣∈G} ={a}.


Conjugate Subgroup

    Let H be a subgroup of a group G. A subgroup K of G is said to be conjugate to H if there exists an element g in G such that K=gHg−1. In other words, K is conjugate to H if K is obtained by conjugating H by an element of G. K is conjugate to H if K={ghg−1∣ℎ∈H} for some g in G.

Quotient Group (Factor Group)

    Suppose N is a normal subgroup of G. The quotient group, denoted as G/N, is formed by partitioning the elements of G into cosets of N and defining a group operation on these cosets.

The elements of G/N are the left cosets of N in G, denoted as gN, where g is any element in G. The group operation is defined by (g1​N) (g2​N) =(g1​g2​) N.

Mathematically, the quotient group G/N is defined as follows:

G/N={gN ∣ g ∈G}

 

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