Stabilizer
A stabilizer is a part of a monoid (or group) acting on a set.
Specifically, let be a monoid operating on a set
, and let
be a subset of
. The stabilizer of
, sometimes denoted
, is the set of elements of
of
for which
; the strict stabilizer' is the set of
for which
. In other words, the stabilizer of
is the transporter of
to itself.
By abuse of language, for an element , the stabilizer of
is called the stabilizer of
.
The stabilizer of any set is evidently a sub-monoid of
, as is the strict stabilizer. Also, if
is an invertible element of
and a member of the strict stabilizer of
, then
is also an element of the strict stabilizer of
, for the restriction of the function
to
is a bijection from
to itself.
It follows that if is a group
, then the strict stabilizer of
is a subgroup of
, since every element of
is a bijection on
, but the stabilizer need not be. For example, let
, with
, and let
. Then the stabilizer of
is the set of nonnegative integers, which is evidently not a group. On the other hand, the strict stabilizer of
is the set
, the trivial group. On the other hand, if
is finite, then the strict stabilizer and the stabilizer are one and the same, since
is bijective, for all
.
Proposition. Let be a group acting on a set
. Then for all
and all
,
.
Proof. Note that for any ,
It follows that
By simultaneously replacing
with
and
with
, we have
whence the desired result.
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