Compatible pair of actions
Contents
Definition
Definition with left action convention
Suppose and
are groups. Suppose
is a homomorphism of groups, defining a group action of
on
. Suppose
is a homomorphism of groups, defining a group action of
on
. For
, denote by
the conjugation map by
. See group acts as automorphisms by conjugation. Then, we say that the actions
form a compatible pair if both these conditions hold:
The above expressions are easier to write down if we use to denote all the actions. In that case, the conditions read:
Here is an equivalent formulation of these two conditions that is more convenient:
-
(the
here is the
of the preceding formulation).
-
(the
here is the
of the preceding formulation)
In the notation, these become:
-
(the
here is the
of the preceding formulation).
-
(the
here is the
of the preceding formulation)
Definition with right action convention
We can give a corresponding definition using the right action convention, but the literature uses the left action convention, so this definition is intended purely as an illustrative exercise. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Symmetry in the definition
Given groups and
with actions
and
,
is compatible with
if and only if
is compatible with
. In other words, the definition of compatibility is symmetric under interchanging the roles of the two groups.
Particular cases
- Trivial pair of actions is compatible: If both the actions are trivial, i.e., both the homomorphisms
are trivial maps, then they form a compatible pair.
- Compatible with trivial action iff image centralizes inner automorphisms: Suppose the homomorphism
is trivial. In that case, the homomorphism
is compatible with
if and only if the image of
in
is in the centralizer
.
- Conjugation actions between subgroups that normalize each other are compatible: If
are both subgroups of some group
that normalize each other (i.e., each is contained in the normalizer of the other), and
are the actions of the groups on each other by conjugation, then they form a compatible pair. Note that in this case, all the actions are just conjugation in
and checking the conditions simply amounts to checking two words to be equal.