Importance of characteristicity
Characteristicity is one of the most important subgroup properties, probably second only to normality in the frequency with which it occurs in group theory.
One of the many basic questions in group theory is the deep similarity as well as contrast in the definitions of normal and characteristic subgroup: a priori, the notion of characteristic subgroup (as being a subgroup invariant under all automorphisms) seems more natural than the notion of normal subgroup (as being a subgroup invariant under only the inner automorphisms). However, it is rather surprising that characteristicity is not as common or important as it may seem -- it is normality that plays the more crucial role (refer importance of normality).
Nonetheless, characteristicity is still fairly important, as we shall explore in this article.
Subgroup-defining functions yield characteristic subgroups
What is a subgroup-defining function?
Further information: subgroup-defining function
A subgroup-defining function is a function that takes in a group and outputs a unique subgroup of that group. For instance:
- The center is the set of those elements that commute with every element
- The commutator subgroup is the subgroup generated by the commutators
- The Frattini subgroup is the intersection of all maximal subgroups
All subgroup-defining functions yield characteristic subgroups
Subgroup-defining functions satisfy the condition of being invariant under isomorphism: any isomorphism of groups, preserves the function. Hence, in particular, the subgroup must be invariant under all automorphisms, and hence, must be a characteristic subgroup.
A natural reverse question is: does every characteristic subgroup arise as the image of some subgroup-defining function? The answer is yes if we are allowed unlimited power in describing the subgroup-defining function. However, if we are restricted to certain kind of languages (for instance, the language of first-order logic, the language of second-order logic, the language of set theory) then we cannot hope to describe every characteristic subgroup of a group.
Importance as the left transiter of normality
One major defect with normality is that it is not transitive. That is, if are subgroups, it is not necessary that . This lack of trnaisitivity of normality often comes in the way of proving results inductively, as well as in trying to use normality at one place to force normality at another.
Characteristic subgroups provide a neat way out. This essentially stems from the fact that every characteristic subgroup of a normal subgroup is normal. In particular, if is normal, then the image of under any subgroup-defining function, is also normal in . Thus, the center of , the commutator subgroup of are all normal in .