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=
The process of finding such a g is called conjugation.
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
Mathematically, the quotient group G/N is defined as follows:
G/N={gN ∣ g ∈G}