Characteristic subgroup

Equivalent definitions in tabular format
Below are many equivalent definitions of characteristic subgroup.

Notation and terminology
For a subgroup $$H \!$$ of a group $$G \!$$, we denote the characteristicity of $$\! H$$ in $$\! G$$ by $$H \operatorname{char} G$$.

Equivalence of definitions
The equivalence of these definitions follows from a more general fact: Restriction of automorphism to subgroup invariant under it and its inverse is automorphism. In other words, we use the fact that both $$\varphi$$ and $$\varphi^{-1}$$ send $$H$$ to within itself to show that $$\varphi(H) = H$$. It is not in general true that if an automorphism of a group restricts to a subgroup, then the restriction is an automorphism of the subgroup: Restriction of automorphism to subgroup not implies automorphism.

Copyable LaTeX
A subgroup $H$ of a group $G$ is termed a {\em characteristic subgroup} if $\varphi(H) = H$ for all automorphisms $\varphi$ of $G$.

Importance
Characteristic subgroups are important because they are genuinely invariant, not just under inner automorphisms, but under all automorphisms. In particular, every subgroup-defining function gives rise to a characteristic subgroup.

Extreme examples

 * 1) Every group is characteristic as a subgroup of itself.
 * 2) The trivial subgroup is characteristic in any group.

Examples using subgroup-defining functions across all groups
In a non-abelian group, some typical examples of characteristic subgroups are given by subgroup-defining functions (something which uniquely returns a particular subgroup). For instance, we have the following subgroup-defining functions, some of which are interesting for abelian groups as well:

For a complete list of subgroup-defining functions, see Category:Subgroup-defining functions.

Similarly, all terms of the upper central series, lower central series, Frattini series, derived series, Fitting series and other series associated with the group, are characteristic.

For a finite group, any normal Sylow subgroup, and more generally, any normal Hall subgroup, is characteristic. More generally, the normal core of any Sylow subgroup or any Hall subgroup, is characteristic.

Metaproperties
Here is a summary:

Relation with other properties
Some of these can be found at:



Stronger properties
The most important stronger properties are fully invariant subgroup (invariant under all endomorphisms) and isomorph-free subgroup (no other isomorphic subgroup).



Conjunction with other properties
Important conjunctions of characteristicity with other subgroup properties (Note that multiple properties listed in the second column indicate that any one of them can be used):



Here are important conjunctions of the property of being a characteristic subgroup with group properties:



In some cases, we are interested in studying characteristic subgroups where the big group is constrained to satisfy some group property. For instance:

Relation with normality

 * Characteristic versus normal: Compares the subgroup properties of characteristicity and normality.
 * Between normal and characteristic and beyond: A survey of the subgroup properties lying between normality and characteristicity..
 * Subnormal-to-normal and normal-to-characteristic: A survey article on subgroup properties such that any normal subgroup satisfying that property is also characteristic.

Varietal formalism
The notion of characteristic subgroup can be defined as the notion of defining ingredient::characteristic subalgebra in the variety of groups.

The second-order description of characteristicity is as follows. We say $$H$$ is characteristic in $$G$$ if:

$$\ \forall g \in H, \sigma \in \operatorname{Aut}(G) : \ \sigma(g) \in H$$

The key point is that quantification over $$\operatorname{Aut}(G)$$ is a second-order quantification.

Characteristicity can be expressed in the relation implication formalism with the left side being automorphs viz subgroups resembling each other via an automorphism of the whole group) and the right side being equal subgroups:

Characteristic = Automorphic subgroups $$\implies$$ Equal subgroups

In other words, a subgroup is characteristic if and only if every subgroup equivalent to it in the sense of being an automorph, is actually equal to it.

The testing problem
Given generating sets for a group and a subgroup, the problem of determining whether the subgroup is characteristic in the group cannot be solved directly. However, it can be reduced to the problem of finding a small generating set for the automorphism group of the bigger group.

The GAP syntax for testing whether a subgroup is characteristic in a group is:

IsCharacteristicSubgroup (group, subgroup);

where subgroup and group may be defined on the spot in terms of generators or may refer to things defined previously.

The list of all characteristic subgroups can be obtained by:

CharacteristicSubgroups(group);

State of discourse
Broadly speaking, "characteristic subgroups" are not a separate area of study in group theory, but rather, characteristic subgroups are an important part of the vocabulary that comes up in any sufficiently "pure" study of group theory. Characteristic subgroups are relatively less important in the study of group actions, and more important in the study of abstract group structure.

History
The notion of characteristic subgroup was introduced by Frobenius in 1895. His motivation was to capture the property of being a subgroup that is invariant under all symmetries of the group, and is hence intrinsic to the group. Frobenius wanted to use the term invariant subgroup but at the time, the term invariant subgroup was used for normal subgroup.

Resolution of questions that are easy to formulate
Any typical question about the behavior of characteristic subgroups in arbitrary groups that is easy to formulate will also be easy to resolve either with a proof or a counterexample, unless some other feature of the question significantly complicates it. This is so, despite the fact that there are a large number of easy-to-formulate questions about the automorphism group that are still open. The reason is that even though not enough is known about the automorphism groups, there are other ways to obtain information about the structure of characteristic subgroups.

At the one extreme, there are abelian groups, where the characteristic subgroups are quite easy to get a handle on. At the other extreme, there are "all groups" where very little can be said about characteristic subgroups beyond what can be proved through elementary reasoning. The most interesting situation is in the middle, for instance, when we are looking at nilpotent groups and solvable groups. In these cases, there are some restrictions on the structure of characteristic subgroups, but the exact nature of the restrictions is hard to work out.

Below are listed some of the more cutting-edge questions that are relatively easy to formulate and are partially open.

Historical references

 * , Page 183
 * : The terminology and language in this book is antiquated, and is only of historical interest. View on Google Books

Textbook references
Advanced undergraduate/beginning graduate algebra texts:

Graduate texts on group theory:

Online lecture notes

 * J.S. Milne's course notes, Section 3.2, Page 41 (both the A4 and the letter version)