Proving that a subgroup is normal

This survey article is about proof techniques for or related to: normality
Find other survey articles about normality

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.

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

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

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.

Normal core

The idea behind using the normal core to prove normality is to show that the given subgroup equals its normal core. In other words, we try to establish that the intersection of all conjugates of the subgroup equals the subgroup itself

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.

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.

Setting up the group action

Let us explore the normal core method a bit further. This method is particularly effective when we are dealing with subgroups of fairly small index.

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. Hence, the quotient group $G/H_{G}$ sits as a subgroup of the symmetric group $Sym(G/H)$.

We may sometimes be able to use this to show that $H=H_{G}$ (using order considerations). For instance, this is the idea behind showing that a subgroup of least prime index is normal.