Normal subgroup

Equivalent definitions in tabular format
Note that each of these definitions (except the first one, as noted) assumes that we already have a group and a subgroup. To prove normality using any of these definitions, we first need to check that we actually have a subgroup.

For more definitions, see nonstandard definitions of normal subgroup.

Notation and terminology
For a subgroup $$H \!$$ of a group $$G \!$$, we denote the normality of $$\! H$$ in $$\! G$$ by $$H \underline{\triangleleft} G$$ or $$G \underline{\triangleright} H$$. In words, we say that $$\! H$$ is normal in $$\! G$$ or a normal subgroup of $$\! G$$.

Copyable LaTeX
The following is LaTeX for a quick definition of normality.

A subgroup $H$ of a group $G$ is termed a {\em normal subgroup} if $ghg^{-1} \in H$ for all $g \in G$ and $h \in H$.

Importance
The notion of normal subgroup is important because of two main reasons:


 * Normal subgroups are precisely the kernels of homomorphisms
 * Normal subgroups are precisely the subgroups invariant under inner automorphisms, and for a group action, the only relevant automorphisms of the acting group that correspond to symmetries of the set being acted upon, are inner automorphisms.

Extreme examples

 * 1) The trivial subgroup is always normal.
 * 2) Every group is normal as a subgroup of itself.

Examples

 * 1) High occurrence example: In an abelian group, every subgroup is normal (there are non-abelian groups, such as the quaternion group, where every subgroup is normal. Groups in which every subgroup is normal are called Dedekind groups, and the non-abelian ones are called Hamiltonian groups).
 * 2) If $$G$$ is an internal direct product of subgroups $$H$$ and $$K$$, both $$H$$ and $$K$$ are normal in $$G$$.
 * 3) Every subgroup-defining function yields a normal subgroup (in fact, it yields a characteristic subgroup). For instance, the center, derived subgroup and Frattini subgroup in any group are normal.

Non-examples
Here are some examples of non-normal subgroups:


 * 1) In the symmetric group on three letters, the subgroup S2 in S3, i.e., the two-element subgroup generated by a transposition, is not normal (in fact, there are three such subgroups and they're all conjugate).
 * 2) More generally, in any dihedral group of degree at least $$3$$, the two-element subgroup generated by a reflection is not normal.
 * 3) Low occurrence example: In a simple group, no proper nontrivial subgroup is normal. Thus, any proper nontrivial subgroup of a simple group gives a counterexample. The smallest simple non-Abelian group is the alternating group on five letters.

Subgroups satisfying the property
Here are examples of subgroups that satisfy the property of being normal:

Subgroups dissatisfying the property
Here are examples of subgroups that do not satisfy the property of being normal.

Relation with other properties
Some of these can be found at:



To get a broad overview, check out the survey articles:


 * Varying normality
 * Subnormal-to-normal and normal-to-characteristic
 * Between normal and characteristic and beyond
 * Between normal and subnormal and beyond
 * Contrasting subnormality of various depths

Stronger properties
The most important stronger property is characteristic subgroup. See the table below for many stronger properties and the way they're related:

Other less important properties that are stronger than normality:

Conjunction with other properties
Important conjunctions of normality with other subgroup properties are in the table below:

 We are often also interested in the conjunction of normality with group properties. By this, we mean the subgroup property of being normal as a subgroup and having the given group property as an abstract group. Examples are in the table below:

 In some cases, we are interested in studying normal subgroups with the big group constrained to satisfy some group property. For instance:

Weaker properties
Other less important properties that are weaker than normality:

Related operators
There are three important subgroup operators related to normality:

Other operators involve composing these in different ways, for instance:


 * Normal core of normalizer
 * Hypernormalizer

Closely related to normal closure is the normal subgroup generated by a subset, which is defined as the smallest normal subgroup containing the subset, and is the normal closure of the subgroup generated by the subset.

Formalisms
The subgroup property of normality can be expressed in first-order language as follows: $$H$$ is normal in $$G$$ if and only if:

$$\forall g \in G, h \in H: \ ghg^{-1} \in H$$

This is in fact a universally quantified expression of Fraisse rank 1.

Normality can be expressed in terms of the relation implication formalism as the relation implication operator with the left side being conjugate subgroups and the right side being equal subgroups:

Conjugate $$\implies$$ Equal

In other words, a subgroup is normal if any subgroup related to it by being conjugate is in fact equal to it.

There are two somewhat different ways of expressing the notion of normality in the language of varieties:


 * In the variety of groups, the normal subgroups are precisely the subalgebras invariant under all the I-automorphisms. An I-automorphism is an automorphism that can be expressed using a formula guaranteed to give an automorphism. This definition of normal subgroup follows from the fact that for groups, inner automorphisms are precisely the I-automorphisms.
 * Treating the variety of groups as a variety of algebras with zero, the normal subgroups are precisely the ideals.

The testing problem
Given generating sets for a group and a subgroup, the problem of determining whether the subgroup is normal in the group reduces to the problem of testing whether the conjugate of any generator of the subgroup by any generator of the group is inside the subgroup. Thus, it reduces to the membership problem for the subgroup.

The GAP syntax for testing whether a subgroup is normal in a group is:

IsNormal (group, subgroup);

where subgroup and group may be defined on the spot in terms of generators (described as permutations) or may refer to things previously defined.

GAP can also be used to list all normal subgroups of a given group, using the command:

NormalSubgroups(group);

Origin of the concept
The notion of normal subgroup dates to an era before group theory began formally. Normal subgroups arose as subgroups for which the quotient group is well-defined.

Normal subgroups were earlier termed invariant subgroups (because they were invariant under inner automorphisms) and also termed self-conjugate subgroups (because a normal subgroup is precisely a subgroup that equals every conjugate).

Origin of the term
The term normal subgroup arose because, under the Galois correspondence established by the fundamental theorem of Galois theory between subgroups and subfields, the normal subgroups corresponded precisely to the subfields that were normal extensions over the base field.

Textbook references
Advanced undergraduate/beginning graduate algebra texts that include group theory:

Graduate texts on group theory:

Online lecture notes

 * J.S. Milne's course notes, Section 1.7, Page 15 (both the A4 and the letter versions)
 * Lecture notes for Benedict Gross's lecture 4, Page 2-3 and more. You can also access the video lectures and other material for the course through the course main page.
 * Lecture notes on basic group theory by James Miller, part of the solitaryroad.com website