Proving that a subgroup is normal

This survey article is about proof techniques for or related to: normal subgroup
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This article explores the various ways in which, given a group and a subgroup (through some kind of description) we can try proving that the subgroup is normal (or that it is not normal). We first discuss the leading general ideas, and then plunge into the specific cases.

Other things instead of proving normality

In some cases, proving that a certain subgroup is normal may be hard or impossible, perhaps because the subgroup is not normal. The following alternative approaches are useful here:

• Replacing a subgroup by a normal subgroup: There are many techniques to guarantee, from the existence of a subgroup satisfying certain conditions, the existence of a normal subgroup satisfying similar conditions.

Using the standard definitions

The best way to try proving that a subgroup is normal is to show that it satisfies one of the standard equivalent definitions of normality.

Methods involving metaproperties of normality

Joins and intersections

Further information: Normality is strongly join-closed,Normality is strongly intersection-closed

If the given subgroup can be described using joins and intersections ,starting with normal subgroups, then it is normal.

Upper joins

Further information: Normality is upper join-closed

If the given subgroup is normal in a bunch of intermediate subgroups that together generate the whole group, it is normal in the whole group.

Quotient-transitivity

Further information: Normality is quotient-transitive

If $H\leq K\leq G$ are such that $H$ is normal in $G$ and $K/H$ is normal in $G/H$, then $K$ is normal in $G$. This is a frequently used fact.

Images, inverse images, transfer, intermediate subgroups

Further information: Normality satisfies intermediate subgroup condition, Normality satisfies transfer condition, Normality satisfies image condition, Normality satisfies inverse image condition

Subgroup-defining function

One of the simplest ways of showing that a subgroup is normal is to show that it arises from a subgroup-defining function. A subgroup-defining function is a rule that associates a unique subgroup to the group.

Any subgroup obtained via a subgroup-defining function is invariant under any automorphism of the group, and is hence a characteristic subgroup. In particular, it is invariant under inner automorphisms of the group, and is hence normal.

With this approach, for instance, we can show that the center, the commutator subgroup, and the Frattini subgroup are normal.

The example of the center: crude versus refined argument

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The deviation method of proving normality

In measuring deviation from normality, we see three ways of measuring the extent to which a subgroup deviates from normality: the normalizer, the normal closure and the normal core. Here, we explore each of these as a tool for trying to prove normality.

The normal core method and group actions

Further information: Group acts on left coset space of subgroup by left multiplication

The idea behind using the normal core to prove normality is to show that the given subgroup equals its normal core: the largest normal subgroup contained in it. In other words, we try to establish that the intersection of all conjugates of the subgroup equals the subgroup itself. This method is particularly useful in cases where the subgroup has small index in the whole group.

The normal core method is typically applied along with group actions, as in the setup described below.

Let $H$ be a subgroup of $G$. Then, $G$ acts on the coset space of $H$. This gives a homomorphism from $G$ to the symmetric group on the coset space, and the kernel of the homomorphism is the normal core $H_{G}$. Hence, the quotient group $G/H_{G}$ sits as a subgroup of the symmetric group $\operatorname {Sym} (G/H)$.

This approach can be used to prove results like the following:

• Index two implies normal
• Subgroup of least prime index is normal
• A related result, that does not prove the subgroup itself is normal, but that it contains another large enough normal subgroup: Poincare's theorem, which states that any subgroup of index $n$ contains a normal subgroup of index dividing $n!$.

Normal closure

The idea behind using the normal closure in order to prove normality is to prove that the subgroup equals its own normal closure. In other words, we show that the subgroup equals that subgroup generated by all its conjugates. This method is particularly useful when the subgroup is given in terms of a generating set.

Suppose $G$ is a group and $H$ is a subgroup with generating set $S$. The normal closure of $H$ in $G$ can be obtained as the subgroup generated by all conjugates of elements of $S$ be elements of $G$. Thus, to show that $H$ is normal in $G$, it suffices to show that all such conjugates are again in $H$.

In fact, if we are given a generating set $A$ for $G$, it suffices to prove that conjugating any element of $S$ by any element in $A\cup A^{-1}$ gives an element of $H$.

Normalizer

The idea behind using the normalizer to prove normality is to prove that the normalizer of the subgroup equals the whole group. In other words, we show that every element of the group commutes with the subgroup.

Methods suited for particular groups

For abelian groups

If the whole group is abelian, then every subgroup is normal, so there is nothing to prove.

For nilpotent groups

• Nilpotent implies every maximal subgroup is normal