Using group actions and representations to solve the extensible automorphisms problem
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An automorphism of a group is termed extensible if, for any group containing , there is an automorphism of whose restriction to equals .
The extensible automorphisms problem asks for a characterization of all the extensible automorphisms of any group. Clearly, any inner automorphism is extensible. The extensible automorphisms conjecture states that the only extensible automorphisms are the inner automorphisms.
Two main methods of attack for the problem are the use of group actions on sets and the use of linear representations, i.e., group actions on vector spaces. This article discusses a general common idea behind these methods, and the reasons behind the partial success as well as the limitations of such approaches.
The main steps of the approach
Step one: Reduce extensibility of the automorphism to being extensible to an inner automorphism with respect to that kind of action
Suppose is a structure with an automorphism group . (The automorphisms are the automorphisms that preserve certain specified structural properties of , these may be all permutations, linear maps, algebra automorphisms, self-homeomorphisms, isometries, or other things). A representation of on is a homomorphism . Two representations and are equivalent if there is an isomorphism such that, for any :
In particular, if , two representations of on are equivalent if there is an automorphism of such that conjugation by that automorphism in sends one homomorphism to the other.
The first step is to use the fact that an automorphism is extensible to deduce that for certain kinds of representations, it can be extended to an inner automorphism of the automorphism group. In other words, we would like to say that if is an extensible automorphism of , then for any representation of on a structure , and are equivalent, i.e., there exists an element of that conjugates to for all .
This is tricky, and the way it is achieved is different in the linear representations and group actions case:
- In the linear representations case, the trick is to consider a semidirect product of the vector space being acted upon with the group, and use the automorphism group action lemma. Further information: Hall-semidirectly extensible implies linearly pushforwardable over prime field
- In the group actions case, the trick is to first do some padding to ensure that the group action is a faithful action, and then, to use the fact that symmetric groups are complete -- they have no outer automorphisms. Further information: Extensible implies permutation-extensible
Step two: relate extensibility to inner automorphisms for every representation to some group-theoretic constraint on the automorphism
Further information: Linearly pushforwardable implies class-preserving for class-separating field, Class-preserving implies linearly pushforwardable, Equivalence of definitions of subgroup-conjugating automorphism
With the completion of step one, we have shown that if is an extensible automorphism of , then can be pushed forward to an inner automorphism of for any homomorphism from to . The next step is to relate the property of being pushforwardable to an inner automorphism of to some intrinsic group-theoretic constraint.
The approach for doing this depends on the specific case. In the case of linear representations, we use conjugacy class-representation duality, that is valid over a sufficiently large field, to show that an automorphism is linearly pushforwardable for all finite-dimensional representations over such a field if and only if it is class-preserving. This duality simply rests on the facts that a representation over a sufficiently large field is completely determined by its character, and any two distinct conjugacy classes can be separated by the character of some representation.
Note that although there is a conjugacy class-representation duality, there is no natural bijection between the conjugacy classes and representations, so transforming the problem from linearly pushforwardability to class-preserving requires us to consider the collection of all irreducible representations.
In the case of permutation representations, we use the fundamental theorem of group actions, that naturally relates conjugacy classes of subgroups with transitive group actions. Using this relation, we can show that an automorphism can be pushed forward for every permutation representation if and only if it is a subgroup-conjugating automorphism, i.e., it sends every subgroup to a conjugate subgroup.