
related topics 
{math, number, function} 
{group, member, jewish} 

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group.
Évariste Galois was the first to realize the importance of the existence of normal subgroups.
Contents
Definitions
A subgroup, N, of a group, G, is called a normal subgroup if it is invariant under conjugation; that is, for each element n in N and each g in G, the element gng^{−1} is still in N. We write
For any subgroup, the following conditions are equivalent to normality. Therefore any one of them may be taken as the definition:
The last condition accounts for some of the importance of normal subgroups; they are a way to internally classify all homomorphisms defined on a group. For example, a nonidentity finite group is simple if and only if it is isomorphic to all of its nonidentity homomorphic images, a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is imperfect if and only if the derived subgroup is not supplemented by any proper normal subgroup.
Full article ▸


related documents 
Compilercompiler 
Identity element 
CYK algorithm 
Toeplitz matrix 
BoxMuller transform 
Hidden Markov model 
De Moivre's formula 
Congruence relation 
Parity (mathematics) 
Nash embedding theorem 
Uncountable set 
Euphoria (programming language) 
Quaternion group 
Deque 
Chomsky normal form 
PSPACEcomplete 
Simple LR parser 
List of logarithmic identities 
Interior (topology) 
Lagrange's theorem (group theory) 
Linear congruential generator 
Divisor 
Binary function 
Event (probability theory) 
Logical disjunction 
Complement (set theory) 
Bézout's theorem 
Sigmaalgebra 
Greedy algorithm 
Homeomorphism 
