Subnormal subgroup

Symbol-free definition
A subgroup of a group is termed subnormal if any of the following equivalent conditions holds:


 * 1) There is a finite ascending chain of subgroups starting from the subgroup and going till the whole group, such that each is a defining ingredient::normal subgroup of its successor. The smallest possible length of such a chain is termed the subnormal depth of the subnormal subgroup.
 * 2) Consider the descending chain of subgroups defined as follows: each member is the normal closure of the original subgroup in its predecessor. This descending chain must reach the original subgroup within finitely many steps. The number of steps it takes is termed the subnormal depth.
 * 3) The sequence of subgroups starting with the whole group, and where each is the commutator of its predecessor with the subgroup, gets inside the given subgroup after finitely many steps. The number of steps is takes is termed the subnormal depth.

The equivalence of the definitions thus must also show that the notions of subnormal depth in each case are the same.

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

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


 * 1) There exists an ascending chain $$H = H_0 \le H_1 \le \dots \le H_n = G$$ 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 $$k$$.

Formalisms
The property of being a subnormal subgroup is obtained by applying the subordination operator to the group property of normality.

Conjunction with other properties

 * Conjunction with subnormal-to-normal subgroup gives normal subgroup
 * Conjunction with pronormal subgroup gives normal subgroup (this generalizes, in fact, to any property that is stronger than being subnormal-to-normal):

In fact, there are a large number of subgroup properties whose conjunction with subnormality gives normality.

Conjunction with group properties:


 * Abelian subnormal subgroup is a subnormal subgroup that is also an abelian group. Also related:
 * Nilpotent subnormal subgroup is a subnormal subgroup that is also a nilpotent group.
 * Cyclic subnormal subgroup is a subnormal subgroup that is also a cyclic group.
 * Solvable subnormal subgroup is a subnormal subgroup that is also a solvable group.
 * Finite subnormal subgroup is a subnormal subgroup that is also a finite group.
 * Perfect subnormal subgroup is a subnormal subgroup that is also a perfect group. Also related:
 * Simple subnormal subgroup is a subnormal subgroup that is also a simple group.
 * Component is a subnormal subgroup that is also a quasisimple group.

The property of being subnormal in particular kinds of groups is also of interest:


 * Subgroup of nilpotent group is a subnormal subgroup of a nilpotent group. (Note that nilpotent implies every subgroup is subnormal).
 * Subnormal subgroup of finite group is a subnormal subgroup of a finite group.
 * Subnormal subgroup of solvable group is a subnormal subgroup of a solvable group.

Related group properties

 * Group in which every subgroup is subnormal is a group in which every subgroup is subnormal. Nilpotent groups have this property: nilpotent implies every subgroup is subnormal.
 * Subnormal intersection property is the property of being a group where an arbitrary intersection of subnormal subgroups is subnormal.
 * Subnormal join property is the property of being a group where a join of finitely many subnormal subgroups is subnormal.
 * Generalized subnormal join property is the property of being a group where an arbitrary join of subnormal subgroups is subnormal.
 * T-group is the property of being a group in which every subnormal subgroup is normal.

Facts

 * There exist subgroups of arbitrarily large subnormal depth: For any positive integer $$k$$, there exists a group $$G$$ and a subgroup $$H$$ of $$G$$ such that the subnormal depth of $$H$$ in $$G$$ is precisely $$k$$..

Metaproperties
More information on these metaproperties:

The property of being a subnormal subgroup is a transitive subgroup property. That is, any subnormal subgroup of a subnormal subgroup is subnormal.

In particular, if $$H \le K \le G$$, with $$H$$ having subnormal depth $$a$$ in $$K$$ and $$K$$ having subnormal depth $$b$$ in $$G$$, the subnormal depth of $$H$$ in $$G$$ is between $$a$$ and $$a + b$$.

The property of being subnormal is trivially true, that is, the trivial subgroup is always subnormal.

The property of being subnormal is also identity-true, that is, every group is subnormal as a subgroup of itself.

The property of being subnormal is a finite-intersection-closed subgroup property. In other words, any finite intersection of subnormal subgroups is a subnormal subgroups.

More generally, an arbitrary intersection of $$k$$-subnormal subgroups is $$k$$-subnormal. The reason why an infinite intersection of subnormal subgroups need not be subnormal is that the subnormal depth of these subgroups need not be bounded. Any arbitrary intersection of subnormal subgroups is a descendant subgroup.

The property of being subnormal is a finite-relative-intersection-closed subgroup property. In other words, if $$H, K \le G$$ are subgroups such that $$H$$ is $$h$$-subnormal in $$G$$ and $$K$$ is $$k$$-subnormal in some subgroup $$L$$ of $$G$$ containing both $$H$$ and $$K$$, then $$H \cap K$$ is $$(h + k)$$-subnormal in $$G$$.

If $$H$$ is a subnormal subgroup of $$G$$, and $$K$$ is an intermediate subgroup of $$G$$ containing $$H$$, then $$H$$ is a subnormal subgroup of $$K$$. In fact, the subnormal depth of $$H$$ in $$K$$ is at most equal to the subnormal depth of $$H$$ in $$G$$.

A join of two subnormal subgroups need not be subnormal. A group where a join of two subnormal subgroups is always subnormal is termed a group satisfying subnormal join property, while a group where a join of arbitrarily many subnormal subgroups is subnormal is termed a group satisfying generalized subnormal join property. All finite groups have these properties; in a finite group, a join of subnormal subgroups is always subnormal.

A subnormal subgroup whose join with any subnormal subgroup is subnormal is termed a join-transitively subnormal subgroup.

Also refer the sections on normalizing joins, permuting joins, and join-transiters.

If $$H, K \le G$$ are subnormal and $$K \le N_G(H)$$, then the join of $$H$$ and $$K$$ is subnormal; in fact, its subnormal depth is bounded by the product of subnormal depths of $$H$$ and $$K$$.

If $$H, K \le G$$ are subnormal and they permute, i.e., $$HK = KH$$, then the product $$HK$$ (which is also the join) is also subnormal. Further, its subnormal depth is bounded by a function of the subnormal depths of $$H$$ and $$K$$.

If $$H$$ is a subnormal subgroup of $$G$$, then for any subgroup $$K \le G$$, the intersection $$H \cap K$$ is a subnormal subgroup of $$K$$. Further, the subnormal depth of $$H \cap K$$ in $$K$$ is bounded by the subnormal depth of $$H$$ in $$G$$.

The centralizer of a subnormal subgroup need not be subnormal.

We can have two subnormal subgroups $$H,K$$ of a group $$G$$ such that $$[H,K]$$ is not a subnormal subgroup.

We can have a subgroup $$H$$ of a group $$G$$ and intermediate subgroups $$K_1, K_2$$ such that $$H$$ is subnormal in both $$K_1$$ and $$K_2$$ but not in the join of subgroups $$\langle K_1, K_2$$. In fact, we can choose an example where $$H$$ is a 2-subnormal subgroup in both $$K_1$$ and $$K_2$$.

Effect of property operators
More on these property operators:

Since subnormality is a t.i. subgroup property, it equals its own left transiter. However, there is a stronger property of being a left-transitively fixed-depth subnormal subgroup.

Since subnormality is a t.i. subgroup property, it equals its own right transiter. However, there is a stronger property of being a right-transitively fixed-depth subnormal subgroup.

A subgroup $$H \le G$$ is termed join-transitively subnormal if $$\langle H, K \rangle$$ is subnormal for any subnormal subgroup $$K$$. Any normal subgroup, 2-subnormal subgroup or permutable subnormal subgroup is join-transitively subnormal.

The testing problem
The property of being subnormal can be tested for permutation groups. In fact, it can be tested if we have an algorithm for normal closure-finding.

The GAP syntax for determining whether a subgroup is subnormal is:

IsSubnormal (Group, Subgroup);

The GAP syntax for finding a subnormal series for a given subgroup is:

SubnormalSeries(Group, Subgroup);

The subnormal depth can be determined by the command:

Length(SubnormalSeries(Group, Subgroup)) - 1;

The -1 is necessary because the length of a series, in GAP, is the number of terms rather than the number of ascent/descent symbols.

Study of the notion
The class 20D35 studies subnormal subgroups for finite groups. In the infinite case, 20E15 is more appropriate.

Textbook references

 * , Page 63 (definition introduced in the context of a more general definition)
 * , Page 92-93 (introduces notion of subnormal series, does not explicitly talk of subnormal subgroup)
 * , Page 23 (formal definition)
 * , Page 13, Chapter 1, Exercise 5 (definition introduced in exercise)

Definition links

 * (stub-length)
 * (stub-length)
 * (stub-length)