Normal subgroup

"Invariant subgroup" redirects here. It is not to be confused with Fully invariant subgroup.

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup H of a group G is normal in G if and only if gH = Hg for all g in G; i.e., the sets of left and right cosets coincide.[1] Normal subgroups (and only 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.[2]


A subgroup, N, of a group, G, is called a normal subgroup if it is invariant under conjugation; that is, the conjugation of an element of N by an element of G is still in N:[3]

NG ⇔ ∀nN, ∀gG, gng−1N.

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 non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images,[4] 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.



Lattice of normal subgroups

The normal subgroups of a group, G, form a lattice under subset inclusion with least element, {e} , and greatest element, G. Given two normal subgroups, N and M, in G, meet is defined as

and join is defined as

The lattice is complete and modular.

Normal subgroups and homomorphisms

If N is normal subgroup, we can define a multiplication on cosets by

(a1N)(a2N) := (a1a2)N.

This turns the set of cosets into a group called the quotient group, G/N. There is a natural homomorphism, f: GG/N given by f(a) = aN. The image f(N) consists only of the identity element of G/N, the coset, eN = N.

In general, a group homomorphism, f: GH sends subgroups of G to subgroups of H. Also, the preimage of any subgroup of H is a subgroup of G. We call the preimage of the trivial group {e} in H the kernel of the homomorphism and denote it by ker(f). As it turns out, the kernel is always normal and the image, f(G), of G is always isomorphic to G/ker(f) (the first isomorphism theorem). In fact, this correspondence is a bijection between the set of all quotient groups, G/N, of G and the set of all homomorphic images of G (up to isomorphism). It is also easy to see that the kernel of the quotient map, f: GG/N, is N itself, so we have shown that the normal subgroups are precisely the kernels of homomorphisms with domain, G.

See also


  1. Thomas Hungerford (2003). Algebra. Graduate Texts in Mathematics. Springer. p. 41.
  2. C.D. Cantrell, Modern Mathematical Methods for Physicists and Engineers. Cambridge University Press, 200, p 160.
  3. Dummit, David S.; Foote, Richard M. (2004), Abstract Algebra (3rd ed.), John Wiley & Sons, ISBN 0-471-43334-9
  4. Pál Dõmõsi and Chrystopher L. Nehaniv, Algebraic Theory of Automata Networks (SIAM Monographs on Discrete Mathematics and Applications, 11), SIAM, 2004, p.7

Further reading

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