Measuring deviation from normality

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Normality is one of the most pivotal subgroup properties. It traces its origins to the very beginnings of group theory, in fact, to even before that. Since a lot of subgroups aren't normal, attempts have naturally been made to determine, given a subgroup, how far it deviates from being normal. Here, we look at some subgroup operators that measure the extent to which a subgroup deviates from normality, aswell as some abstract groups we can associate with the given subgroup that again measure the deviation.

Subgroup operators measuring the deviation

Normal closure

Further information: normal closure

Recall the following important fact:

An arbitrary intersection of normal subgroups is a normal subgroup. In other words, normality is an intersection-closed subgroup property.

Thus, given a subgroup $H \le G$ we can look at the set of all normal subgroups of $G$ containing $H$, and take their intersection. This intersection will itself be a normal subgroup, and it will contain $H$. The intersection is termed the normal closure of $H$. The normal closure of $H$ in $G$ is denoted as $H^G$.

A subgroup is normal if and only if it equals its normal closure.

The normal closure of $H$ can be viewed as a normal approximation to it from above, that is, a normal approximation that is bigger than it.

Another way of viewing the normal closure is as the smallest subgroup that contains $H$ as well as all the conjugates of $H$. This can be thought of as a constructive way of making the normal closure, starting from $H$ and moving upwards (as opposed to the intersection route that goes downwards).

Normal core

Further information: normal core

Recall the following important fact:

An arbitrary join of normal subgroups is a normal subgroup. In other words, normality is a join-closed subgroup property.

Thus, given a subgroup $H \le G$, we can take the join of all normal subgroups of $G$ contained inside $H$. The join will itself be a normal subgroup, and it will contain $H$. This join is termed the normal core of $H$ in $G$. The normal core of $H$ in $G$ is denoted as $H_G$.

A subgroup is normal if and only if it equals its normal core.

The normal core of a subgroup can be viewed as a normal approximation to it from below, that is, a normal approximation that is smaller than it.

Another way of viewing the normal core is as the intersection of $H$ and all the conjugates of $H$.

Normalizer

Further information: normalizer

Recall the following definition of normality: $H \le G$ is normal if and only if $xH = Hx$ for every $x \in G$.

The normalizer of a subgroup is defined as the set of all elements that commute with the subgroup. In other words, the normalizer of a subgroup $H \le G$, denoted as $N_G(H)$, is defined as the set of all $x$ such that $xH = Hx$.

Clearly, $H$ is normal in $N_G(H)$, and any subgroup in which $H$ is normal must lie inside $N_G(H)$. Thus, $N_G(H)$ is the largest subgroup in which $H$ is normal.

A subgroup is normal if and only if its normalizer equals the whole group.

Opposites of normality via this deviation

Contranormal subgroup

Further information: contranormal subgroup

A subgroup of a group is termed contranormal if its normal closure is the whole group. In other words, the subgorup generated by all its conjugates is the whole group, or equivalently, there is no proper normal subgroup containing it.

The property of being contranormal is a NCI-subgroup property, that is, no proper normal subgroup can be contranormal. In fact, no proper subnormal subgroup can be contranormal either, because if $H = H_0 \triangleleft H_1 \triangleleft \ldots \triangleleft H_{n-1} \triangleleft H_n = G$ is a subnormal series then $H_{n-1}$ is a proper normal subgroup containing $H$.

Note that the condition of being contranormal does not imply that every element of the group is conjugate to some element in the subgroup, rather, it says that every element in the group can be expressed as a product of elements in the subgroup and their conjugates. The stronger condition of every element in the group being conjugate to some element in the subgroup, is termed the condition of being conjugate-dense.

Further, it is true that in a simple group, every nontrivial subgroup is contranormal, because its normal closure must be a nontrivial normal subgroup, and hence the whole group.

Core-free subgroup

Further information: core-free subgroup

A subgroup of a group is termed core-free if its normal core is the trivial subgroup. In other words, the intersection of all its conjugates is trivial, or equivalently, there is no nontrivial normal subgroup contained inside it.

The property of being core-free is a NCT-subgroup property, that is, no nontrivial normal subgroup can be core-free. It may, however, happen that there is a subnormal core-free subgroup.

It is also true that in a simple group, any proper subgroup is core-free. This follows from the fact that the normal core of any proper subgroup must be a proper normal subgroup, and hence, trivial.

There are a number of stronger conditions than being core-free. For instance, there is the condition of being a malnormal subgroup, viz a subgroup such that the intersection with any conjugate by an outside element is trivial. (Any proper malnormal subgroup is core-free).

Self-normalizing subgroup

Further information: self-normalizing subgroup

A subgroup of a group is termed self-normalizing if it equals its own normalizer. In other words, a subgroup of a group is termed self-normalizing if there is no subgroup properly containing it in which it is normal.

The property of being self-normalizing is a NCI-subgroup property, that is, no proper normal subgroup can be self-normalizing. In fact, no proper subnormal subgroup can be self-normalizing, because if $H = H_0 \triangleleft H_1 \triangleleft \ldots \triangleleft H_n = G$ is a subnormal series for $H$, then $H$ is normal in $H_1$.

Unlike the other two cases (viz contranormal and core-free subgroup) it is not true that most subgroups of a simple group are self-normalizing. This can essentially be viewed as because of the fact that the normalizer is local more to the nature of the subgroup than to the nature of the whole group.

Comparison between all the three properties

The properties of being self-normalizing, core-free and contranormal are all closely related, thoguh none of them implies any of the others. Here are some facts.

Properties which imply both self-normalizing and core-free:

• Proper malnormal subgroup

Properties which imply both self-normalizing and contranormal:

• Maximal subgroup which is not normal
• Abnormal subgroup
• Weakly abnormal subgroup

Properties which imply both contranormal and core-free:

• Strongly contranormal subgroup
• Conjugate-lattice-complemented subgroup

Iteration of these operators and factorization

Contranormal-descendant factorization

Given a subgroup $H \le G$, consider the sequence $G_\alpha$ of subgroups defined as follows:

• $G_0 = G$
• $G_{\alpha + 1}$ is the normal closure of $H$ in $G_{\alpha}$ for any ordinal $\alpha$
• $G_\alpha = \bigcap_{\beta < \alpha} G_\beta$ if $\alpha$ is a limit ordinal

This is a descending series of subgroups and must eventually stabilize at some ordinal $\alpha$. Clearly, we will then have the following:

• $G_\alpha$ is a descendant subgroup of $G$ (in finite groups, this is the same as saying that $G_\alpha$ is a subnormal subgroup of $G$)
• $H$ is a contranormal subgroup of $G_\alpha$

Moreover, $G_\alpha$ is the only intermediate subgroup for which both these conditions hold.

Normal-core-free factorization

Given a subgroup $H \le G$, we have:

• $H_G$ is a normal subgroup of $G$ contained inside $H$
• $H/H_G$ is a core-free subgroup of $G/H_G$

Moreover $H_G$ is the unique subgroup for which both these hold.

Ascendant-self-normalizing factorization

Given a subgroup $H \le G$, we can find an intermediate subgroup $K$ such that:

• $H$ is an ascendant subgroup of $K$ (in finite groups, this is the same as requiring that $H$ is a subnormal subgroup of $K$)
• $K$ is a self-normalizing subgroup of $G$

Abstract groups measuring deviation from normality

Quotient-by-core

Further information: quotient-by-core operator

Given a subgroup $H \le G$, we consider the group $H/H_G$ where $H_G$ denotes the normal core of $H$. Note that this group is trivial when $H$ is normal, so the size of this group measures the extent to which the subgroup deviates from normality.

For a core-free subgroup, the quotient-by-core is the same as the subgroup.

Certain interesting results are known on the quotient by the normal core. For instance, the quotient-by-core of a permutable subgroup, and more generally, of a modular subgroup in a finite group, is a nilpotent group.

Closure-by-core

Further information: closure-by-core operator

Given a subgroup $H \le G$, the closure-by-core operator gives the abstract group $H^G/H_G$, where $H^G$ is the normal closure of $H$ in $G$ and $H_G$ is the normal core of $H$ in $G$. In other words, the closure-by-core operator outputs the quotient of the normal closure by the normal core.

Note that, like the quotient-by-core, this group is trivial when the subgroup $H$ is normal, so the size of this group measures deviation from normality.

Normalizer-by-self

Further information: normalizer-by-self operator

Given a subgroup $H \le G$, the normalizer-by-self operator gives the abstract group $N_G(H)/H$. Note that this is well-defined because $H$ is normal in its normalizer.

The normalizer-by-self of a normal subgroup is the corresponding quotient group.