Contrasting subnormality of various depths

This survey article compares, and contrasts, the following subgroup properties: normal subgroup versus subnormal subgroup
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This survey article compares the property of being a normal subgroup, the property of being a subnormal subgroup, and the property of being a subgroup of subnormal depth at most $k$: in other words, the property of being a $k$-subnormal subgroup. The particular cases of interest are:

• $k=0$: This is the property of being the whole group.
• $k=1$: This is the property of being a normal subgroup.
• $k=2$: This is the property of being a 2-subnormal subgroup.
• $k=3$: This is the property of being a 3-subnormal subgroup.
• $k=4$: This is the property of being a 4-subnormal subgroup.

It turns out that the bulk of special behavior occurs for small values of $k$.

Definitions

Normal subgroup

Further information: normal subgroup

A subgroup $N\!$ of a group $G\!$ is said to be normal in $G$ (in symbols, $N\triangleleft G$ or $G\triangleright N$^Notations) if the following equivalent conditions hold:

1. (Homomorphism kernel definition): There is a homomorphism $\varphi$ from $G$ to another group such that the kernel of $\varphi$ is precisely $N\!$.
2. (Inner automorphisms definition): For all $g\in G$, $gNg^{-1}\subseteq N$. More explicitly, for all $g\in G,h\in N$, we have $ghg^{-1}\in N$.
3. (Equals conjuate definition): For all $g$ in $G$, $gNg^{-1}=N$.
4. (Cosets definition): For all $g$ in $G$, $gN=Ng$.
5. (Conjugacy classes definition): $N$ is a union of conjugacy classes.
6. (Commutator definition): The commutator $[N,G]$ is contained in $N$.

Subnormal subgroup

Further information: Subnormal subgroup

A subgroup $H$ is termed subnormal in a group $G$ if either of the following equivalent conditions holds:

1. There exists an ascending chain $H=H_{0}\leq H_{1}\dots H_{n}=G$ such that each $H_{i}$ is normal in $H_{i+1}$. The smallest possible $n$ for which such a chain exists is termed the subnormal depth of $H$.
2. Consider the descending chain $G_{i}$ defined as follows: $G_{0}=G$ and $G_{i+1}$ is the normal closure of $H$ in $G_{i}$. Then, there exists an $n$ for which $G_{n}=H$. The smallest such $n$ is termed the subnormal depth of $H$.
3. Consider the sequence $K_{i}$ of subgroups of $G$ defined as follows: $K_{0}=G$, and $K_{i+1}=[H,K_{i}]$ (the commutator), This sequence of subgroups eventually enters inside $H$. The number of steps taken is termed the subnormal depth of $H$.

A $k$-subnormal subgroup is a subnormal subgroup with subnormal depth at most $k$.

The lack of transitivity for normality, and the birth of subnormality

Normality is not transitive

Further information: Normality is not transitive

A normal subgroup of a normal subgroup need not be normal. In other words, we can have groups $H\leq K\leq G$ such that $H$ is normal in $K$ and $K$ is normal in $G$, but $H$ is not normal in $G$. The smallest counterexample group is dihedral group:D8.

The property of subnormality can be viewed as a remedy to the lack of transitivity of normality. A subnormal subgroup is a subgroup that can be obtained by taking a normal subgroup of a normal subgroup of a normal subgroup ... done finitely many times. The property of subnormality can also be viewed as obtained by applying the subordination operator to the property of normality.

There exist subgroup of arbitrarily large subnormal depth

Further information: There exist subgroups of arbitrarily large subnormal depth

The fact that normality is not transitive yields that 2-subnormality, i.e., the property of being a normal subgroup of a normal subgroup, is a strictly weaker condition than normality. In fact, something stronger is true: $(k+1)$-subnormality is strictly weaker than $k$-subnormality for every $k$. In other words, there exist subgroups of arbitrarily large subnormal depth.

Ascendant, descendant, and serial subgroups

Further information: Ascendant subgroup, descendant subgroup, serial subgroup, ascendant not implies subnormal, descendant not implies subnormal

For subnormality, we require a finite chain of subgroups between the subgroup and the whole group, with each member normal in its successor. Some variations on this theme are:

• Ascendant subgroup, which has a transfinite well-ordered ascending series with each member normal in its successor, and the union of all members before a limit ordinal is normal in the member corresponding to that limit ordinal.
• Descendant subgroup, which is the analogous notion for a well-ordered descending series.
• Serial subgroup, where there is a totally ordered chain, which need not be well-ordered in either direction, but for which the union of members to the left of any cut is normal in the intersection of members to the right.

Playing with transitivity for a fixed subnormal depth

Subnormality is transitive, but subnormality of fixed depth is not

A subnormal subgroup of a subnormal subgroup is subnormal. In particular, a subnormal subgroup of a normal subgroup is subnormal, and a normal subgroup of a subnormal subgroup is subnormal.

Specifically, a $k$-subnormal subgroup of a $l$-subnormal subgroup is $(k+l)$-subnormal. However, we cannot in general guarantee any better bound on the subnormal depth, because there exist subgroups of arbitrarily large subnormal depth, as pointed earlier.

In particular, for $k\geq 1$, the property of $k$-subnormality is not transitive.

General definition of left and right transiter

Further information: Left transiter, right transiter

Let $p$ be a subgroup property. The left transiter of $p$ is defined as the following property $q$: a subgroup $H$ of a group $K$ has property $q$ in $K$ if whenever $K\leq G$ is such that $K$ has property $p$ in $G$, $H$ also has property $p$ in $G$.

The right transiter if $p$ is defined as the following property $q$: a subgroup $K$ of a group $G$ has property $q$ in $G$ if whenever $H\leq K$ is such that $H$ has property $p$ in $K$, $H$ has property $p$ in $G$.

A t.i. subgroup property is a subgroup property $p$ that is both identity=true (every group has the property as a subgroup of itself) and transitive. $p$ is transitive if whenever $H\leq K\leq G$ such that $H$ satisfies property $p$ in $K$ and $K$ satisfies property $p$ in $G$, $H$ also satisfies property $p$ in $G$.

As discussed above, subnormality is transitive, and since every group is subnormal in itself, it is t.i.. It turns out that the left transiter and right transiter of any subgroup property are t.i., and conversely, if a subgroup property is t.i., it equals both its left and right transiter.

Relation between left and right transiters for $k$-subnormality for different values of $k$

Further information: Left-transitively fixed-depth subnormal subgroup, Normal not implies left-transitively fixed-depth subnormal, Right-transitively fixed-depth subnormal subgroup, Normal not implies right-transitively fixed-depth subnormal

Although subnormality is transitive, $k$-subnormality for fixed $k$ is not. Thus, it is interesting to study the left and right transiters of $k$-subnormality. Two definitions:

• The left transiter of $k$-subnormal is termed left-transitively $k$-subnormal. Further, if $k\leq l$, any left-transitively $k$-subnormal subgroup is left-transitively $l$-subnormal. A subgroup that is left-transitively $k$-subnormal for some $k\geq 1$ is termed a left-transitively fixed-depth subnormal subgroup. It turns out that not every normal subgroup, and hence not every subnormal subgroup, is left-transitively fixed-depth subnormal.
• The right transiter of $k$-subnormal is termed right-transitively $k$-subnormal. Further, if $k\leq l$, any right-transitively $k$-subnormal subgroup is right-transitively $l$-subnormal.A subgroup that is right-transitively $k$-subnormal for some $k\geq 1$ is termed a right-transitively fixed-depth subnormal subgroup. It turns out that not every normal subgroup, and hence not every subnormal subgroup, is right-transitively fixed-depth subnormal.

Left and right transiters for normality

Further information: Characteristic of normal implies normal, Left transiter of normal is characteristic, transitively normal subgroup, central factor implies transitively normal

The left transiter of normality is the property of being a characteristic subgroup. A subgroup $H$ of a group $K$ is termed characteristic in $K$ if $H$ is invariant under all automorphisms of $K$.

In other words, $H\leq K$ is characteristic if and only if whenever $K$ is normal in $G$, so is $H$.

The right transiter of normality is termed the property of being a transitively normal subgroup. There are a number of properties that imply being transitively normal. For instance, $H$ is a central factor of $G$ if $HC_{G}(H)=G$, or equivalently, every inner automorphism of $G$ restricts to an inner automorphism of $H$. Any central factor is transitively normal. In particular, any central subgroup, cocentral subgroup, or direct factor is transitively normal.

Right transiters for 2-subnormality

Further information: Right-transitively 2-subnormal subgroup, Base of a wreath product implies right-transitively 2-subnormal, Transitively normal implies right-transitively 2-subnormal

The right transiter of the property of being a 2-subnormal subgroup is termed right-transitively 2-subnormal subgroup. By the previous discussion, a transitively normal subgroup is right-transitively 1-subnormal, and hence right-transitively 2-subnormal. However, there are many subgroups that are not transitively normal, and are in fact not even normal, but are right-transitively 2-subnormal.

For instance, a base of a wreath product is right-transitively 2-subnormal. Similarly, any abelian normal subgroup, Dedekind normal subgroup, or normal T-subgroup is right-transitively 2-subnormal.

Left transiters for 2-subnormality

Further information: Left-transitively 2-subnormal subgroup, Subgroup-cofactorial automorphism-invariant implies left-transitively 2-subnormal

The left transiter of the property of being a 2-subnormal subgroup is termed left-transitively 2-subnormal subgroup. By the previous discussion, any characteristic subgroup is left-transitively 2-subnormal. However, a left-transitively 2-subnormal subgroup is not necessarily characteristic. In fact, any subgroup-cofactorial automorphism-invariant subgroup is left-transitively 2-subnormal: a finite subgroup $H$ of a group $G$ is termed subgroup-cofactorial automorphism-invariant if, for any automorphism $\sigma$ of $G$ such that every prime factor of the order of $\sigma$ divides the order of $H$, $H$ is invariant under $\sigma$. (There are stronger versions of this result, where we require invariance only under automorphisms that have order equal to the order of some element of the subgroup).

Some general comments

As remarked earlier, the property of being left-transitively $k$-subnormal becomes successively weaker as $k$ gets larger. The cases $k=1$ and $k=2$ are the most interesting and clearly understood, with there being important properties that are stronger than the property of being left-transitively 2-subnormal. Similar remarks hold for right-transitively $k$-subnormal. This is typical for subnormality: the really interesting behavior is for normal subgroups and 2-subnormal subgroups, and the behavior for specific higher values of $k$ is not as interesting.

Playing with intersections

Closure under arbitrary intersections for fixed depth

Further information: Subnormality of fixed depth is strongly intersection-closed, Normality is strongly UL-intersection-closed, Subnormality is strongly UL-intersection-closed

An arbitrary intersection of $k$-subnormal subgroups is $k$-subnormal. Note that this does not automatically follow from the fact that an arbitrary intersection of normal subgroups is normal. Rather, it uses the stronger fact that normality is strongly UL-intersection-closed: If $H_{i}\leq K_{i}\leq G$ are such that each $H_{i}$ is normal in $K_{i}$, then the intersection of the $H_{i}$s is normal in the intersection of the $K_{i}$s.

Not closed under arbitrary intersections for variable depth

Further information: Descendant not implies subnormal

An arbitrary intersection of subnormal subgroups need not be subnormal. This is related to the fact that descendant subgroups are not subnormal. Note that a subgroup is an intersection of subnormal subgroups if and only if it is either subnormal or descendant with a descendant series of length $\omega$.

Intermediate subgroups, transfer condition, images and inverse images

We have the following:

• Subnormality satisfies intermediate subgroup condition: In fact, if $H\leq K\leq G$ and $H$ is $k$-subnormal in $G$, then $H$ is $k$-subnormal in $K$.
• Subnormality satisfies transfer condition: In fact, if $H,K\leq G$ and $H$ is $k$-subnormal in $G$, then $H\cap K$ is $k$-subnormal in $K$.
• Subnormality satisfies image condition: In fact, if $\varphi :G\to K$ is a surjective homomorphism and $H$ is a $k$-subnormal subgroup of $G$, $\varphi (H)$ is also a $k$-subnormal subgroup of $K$.
• Subnormality satisfies inverse image condition: The inverse image of a $k$-subnormal subgroup under any homomorphism of groups is also a $k$-subnormal subgroup.

Playing with joins

Not in general closed under finite joins

Further information: Subnormality is not finite-join-closed, Join of two 3-subnormal subgroups may be proper and contranormal

A join of two subnormal subgroups need not be subnormal. Counterexamples are tricky to construct because, as discussed below, if either of the subgroups is a 2-subnormal subgroup, or if the whole group satisfies any of a large number of hypotheses such as being finite, nilpotent, or slender, the join is subnormal.

Some preliminary results

Further information: Join of normal and subnormal implies subnormal of same depth, Subnormality is normalizing join-closed, Subnormality is permuting join-closed

We have the following results:

• Join of normal and subnormal implies subnormal of same depth: If $H$ is a normal subgroup of $G$ and $K$ is a $k$-subnormal subgroup of $G$, the join $\langle H,K\rangle$, which in this case equals the product $HK$, is also $k$-subnormal.
• Subnormality is normalizing join-closed: If $H,K\leq G$ are subnormal subgroups such that $K\leq N_{G}(H)$, the join is subnormal, and its subnormal depth is at most equal to the product of the subnormal depths of $H$ and $K$.
• Subnormality is permuting join-closed: If $H,K\leq G$ are subnormal subgroups such that $HK=KH$, i.e., they permute, the join is subnormal, and its subnormal depth is bounded as a function of the subnormal depths of $H$ and $K$.

The relation between join, commutator and closure under conjugation

Further information: Join of subnormal subgroups is subnormal iff their commutator is subnormal

One of the crucial results about joins is the following: If $H,K\leq G$ are subnormal subgroups, the join $\langle H,K$ is subnormal if and only if the commutator $[H,K]$ is subnormal, if and only if $H^{K}$ is subnormal. Further, the subnormal depths of these three subgroups are bounded in terms of each other and the subnormal depths of $H$ and $K$.

The relation between joins, conjugate joins, automorph joins, and other joins

Further information: Conjugate-join-closed subnormal implies join-transitively subnormal, Join-transitively subnormal implies finite-automorph-join-closed subnormal, Finite-automorph-join-closed subnormal of normal implies finite-conjugate-join-closed subnormal

A subgroup $H$ of a group $G$ is termed a conjugate-join-closed subnormal subgroup if a join of an arbitrary number of conjugate subgroups of $H$ is subnormal. We similarly define finite-conjugate-join-closed subnormal subgroup and finite-automorph-join-closed subnormal subgroup.$H$ is termed join-transitively subnormal if its join with any subnormal subgroup is subnormal.

We have the following easy-to-deduce results:

• Conjugate-join-closed subnormal implies join-transitively subnormal: This follows from the result of the previous subsection.
• Join-transitively subnormal implies finite-automorph-join-closed subnormal
• Automorph-join-closed subnormal of normal implies conjugate-join-closed subnormal
• Finite-automorph-join-closed subnormal of normal implies finite-conjugate-join-closed subnormal

These facts together yield that for finite groups, a join of two subnormal subgroups is subnormal.

The special case of normal subgroups

Further information: Normality is strongly join-closed, Join of normal and subnormal implies subnormal of same depth

The special behavior of 2-subnormal subgroups

Further information: 2-subnormality is conjugate-join-closed, 2-subnormal implies conjugate-join-closed subnormal, 2-subnormal implies join-transitively subnormal

Using the ideas of the previous subsection, we can show that an arbitrary join of conjugate 2-subnormal subgroups is 2-subnormal. This can be used to show that the join of a $2$-subnormal subgroup and a $k$-subnormal subgroup is $2k$-subnormal.

The special behavior of 3-subnormal subgroups

Further information: 3-subnormal implies finite-conjugate-join-closed subnormal, Join of 3-subnormal and finite subnormal implies subnormal

A join of finitely many conjugate $3$-subnormal subgroups is still subnormal, and its subnormal depth can be bounded in terms of the number of subgroups joined. Further, the join of a $3$-subnormal subgroup and any finite subnormal subgroup is again subnormal, with the depth in terms of the order of the finite subnormal subgroup.

Other remarks about subnormal joins

In general, the lower the subnormal depth, the more guarantees we can make about the behavior with respect to joins.

Upper joins and existence of subnormalizers

For normal subgroups

Further information: Normality is upper join-closed

If $H$ is a subgroup of $G$ and $K_{i},i\in I$ is a collection of subgroups of $G$ containing $H$, such that $H$ is normal in each $K_{i}$, then $H$ is also normal in the join of the $K_{i}$s.

In fact, there is a unique largest subgroup of $G$ in which $H$ is normal, and this subgroup is termed the normalizer of $H$ in $G$.

For 2-subnormal subgroups

Further information: 2-subnormality is not upper join-closed

For subnormal subgroups in general

Further information: Subnormality is not upper join-closed

Suppose $H$ is a subgroup of a group $G$. A subnormalizer of $H$ is a subgroup $K$ of $G$ such that $K$ is the unique largest subgroup of $G$ in which $H$ is subnormal. If $H$ has a subnormalizer, it is termed a subgroup having a subnormalizer. Note that any subnormal subgroup has a subnormalizer, and there are many other subgroup properties, such as being an intermediately subnormal-to-normal subgroup, that imply the existence of a subnormalizer.

A slightly different notion, that we call the subnormalizer subset (though this is also called the subnormalizer at times) is the set of all elements $g\in G$ such that $H$ is subnormal in $\langle H,g\rangle$. Note that this set of elements need not be a subgroup, and even if it is a subgroup, it could potentially be a lot bigger than the subnormalizer subgroup of $H$, even if the latter does exist.