Questions:Normal subgroup

This is a questions page on normal subgroup, listing common questions that may occur both to people encountering the definition for the first time and to people who have spent some time with the definition.

In addition to reading these questions, you should read: normal subgroup (the main page on the topic), nonstandard definitions of normal subgroup, proving normality, using normality, normal versus characteristic, ubiquity of normality.

Also see facts about normality, facts that prove that a subgroup is normal, and facts that use that a subgroup is normal.

Q: There are so many different definitions of normal subgroup. Which of these is the correct definition? Which one should I use in practical situations?

A: The two most important definitions of normality are the definition as the kernel of a homomorphism of groups and the definition in terms of invariance under inner automorphisms. However, the other definitions are also important and worth knowing. What's really important about normality is not any one of these definitions per se, but the remarkable fact that all these definitions are equivalent. To see how different definitions can be used, both for proving that a subgroup is normal and for using that a subgroup is normal, see proving normality and using normality.

Q: The definition of normality seems to be somewhat related to the notion of abelian group. What precisely is the relation?

A: The main relation is abelian implies every subgroup is normal: in an abelian group, every subgroup is normal. This is best seen from the conjugation/inner automorphism definition, because in an abelian group, every element is invariant under conjugation by every other element. It can also be seen from the left coset/right coset definition or the commutator definition. Interestingly, there do exist non-abelian groups in which every subgroup is normal, such as the quaternion group. Further information: Dedekind not implies abelian

It is not necessary that an abelian subgroup of a non-abelian group be normal. The easiest counterexample is the subgroups of order two in the symmetric group of degree three (see S2 is not normal in S3). It is also not true that any subgroup of an abelian normal subgroup is normal. An example is the dihedral group:D8, which has an abelian normal subgroups of order four (the Klein four-subgroups) which in turn have subgroups that are not normal in the whole group.

Q: I also learned something about the center. What is the difference between the notion of center and the notion of normal subgroup?

A: The center is defined as the set of elements that are, as individual elements, invariant under conjugation by other elements. On the other hand, a normal subgroup has to be invariant under conjugation as a set -- conjugation may move elements within the set.

It is true that the center is normal. More generally, a central subgroup is a subgroup of the center, and any central subgroup is normal.

However, every normal subgroup need not be central. In fact, even an abelian normal subgroup need not be central. Examples include the normal subgroup of order three in the symmetric group of degree three and the normal subgroups of order four in the dihedral group:D8.

Q: One definition I saw said that $N$ is normal in $G$ if $gNg^{-1}\leq N$ for all $g\in G$. Another definition used $gNg^{-1}=N$. Why are these equivalent?

A: This is a very good and somewhat tricky question. The $=$ formulation seems stronger, but it turns out to be equivalent. Interestingly, it is not true that if, for a particular element $g$ and a subgroup $N$, $gNg^{-1}\leq N$, then $gNg^{-1}=N$. Rather, it is the fact that this is true for every $g$ that matters.

This is because restriction of automorphism to subgroup invariant under it and its inverse is automorphism. Now, if $N$ is normal in $G$ by the $\leq$ definition, we not only have $gNg^{-1}\leq N$, we also have that $g^{-1}Ng\leq N$. Thus, $N$ is invariant both under conjugation by $g$ and under conjugation by $g^{-1}$, which are inverse operations of each other. This forces $gNg^{-1}=N$. (To see this without using the fact quoted above, note that $g^{-1}Ng\leq N$, implies, via left multiplication by $g$ and right multiplication by $g^{-1}$, that $N\leq gNg^{-1}$. Combined with $gNg^{-1}\leq N$, we obtain $N=gNg^{-1}$).

Note also that the use of the $\leq$ symbol, as opposed to the $\subseteq$ symbol, already encodes the information that conjugation by $g$ is an automorphism, hence the image of $N$ is a subgroup. If you read this definition before these ideas were introduced, you may have seen the notation $gNg^{-1}\subseteq N$. The same ideas apply with the set notation.

Q: I was told that it is very important and counterintuitive that a normal subgroup of a normal subgroup need not be normal, but I didn't find it either important or counterintuitive. What is the significance?

A: You're referring to the fact that normality is not transitive. This is indeed important, though it is not necessarily counterintuitive. The importance is partly because if the opposite were true, it would prove a very convenient way of showing that subgroups are normal, and thus make group theory very different and perhaps more boring.

One way of thinking about the significance is to look at the implications for quotient groups. If $H\leq K\leq G$ with $H$ normal in $K$ and $K$ normal in $G$, we can talk of the quotient groups $G/K$ and $K/H$. If it were also true that $H$ is normal in $G$, we'd have a group $G/H$, whereby $K/H$ could be identified with a normal subgroup of it and the quotient would be isomorphic to $G/K$ (by the third isomorphism theorem). However, the point is that we are not guaranteed that $H$ is normal in $G$. This fact shows that there is, in some sense, a lot more flexibility in the way groups can be put together.

Q: I have seen the definition of normality as invariance under conjugations, but then read something about these conjugations also being called inner automorphisms. Does this have any significance?

A: That the conjugation operations are inner automorphisms, and that invariance under these is equivalent to being normal, is very significant. This is because inner automorphisms are automorphisms, so they preserve the group structure. This is one explanation for why normality is strongly join-closed (the subgroup generated by a bunch of normal subgroups is normal), normality is centralizer-closed (the centralizer of a normal subgroup is normal), and normality is commutator-closed (the commutator of two normal subgroups is normal). It also explains why characteristic subgroups are normal, which explains why subgroup-defining functions such as the center, commutator subgroup, socle, and Frattini subgroup are all normal subgroups.

Q: Could you clarify the relationship between normal subgroups and characteristic subgroups?

A: Characteristic means invariant under all automorphisms, normal means invariant under inner automorphisms. Characteristic implies normal, normal not implies characteristic, and characteristic of normal implies normal. See more at characteristic versus normal.

Q: I was told by some people that normal subgroups are precisely the subgroups that are invariant, but the correct term for that seems to be characteristic subgroup. How do you explain this?

A: It depends on what sort of invariance is being sought. If we are looking for invariance under automorphisms of the group, then the correct notion is that of characteristic subgroup.

However, normality is more important in the following sense: when a group acts on a structure, we are interested in those subgroups of the group that are invariant under change of coordinates on the set, which means invariant under automorphisms of the set. Since the group itself acts by automorphisms, this in particular implies invariance under the action of the group. But when a group acts on a set, the induced action on itself is the group action by conjugation. Thus, all subgroups invariant in this manner must be normal; however, they need not be characteristic. Since most of the applications of groups to other areas of mathematics is via group actions, normality is the more important notion. Further information: Ubiquity of normality

Q: The product of a normal subgroup with any subgroup is a subgroup. Are normal subgroups the only subgroups with this property?

A: No. A subgroup whose product with any subgroup is a subgroup is termed a permutable subgroup (or quasinormal subgroup), i.e., it permutes with every subgroup. normal subgroups are permutable but permutable not implies normal. For instance, any subgroup of the Baer norm (the intersection of normalizers of all subgroups) is permutable but it need not be normal.