Difference between revisions of "Fixer"
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− | A '''fixer''' is part of a [[monoid]] (or [[group]]) acting on a [[set]]. | + | A '''fixer''' is part of a [[monoid]] (or [[group]]) [[group action|acting]] on a [[set]]. |
Specifically, let <math>M</math> be a monoid acting on <math>S</math>; let <math>A</math> be a subset of <math>S</math>. The fixer of <math>S</math> is the set of all <math>a\in M</math> fow which <math>a(x) = x</math> for all <math>x \in A</math>. If <math>S</math> contains a single element <math>x</math>, we sometimes call this the fixer or [[stabilizer]] of <math>x</math>, by abuse of language. | Specifically, let <math>M</math> be a monoid acting on <math>S</math>; let <math>A</math> be a subset of <math>S</math>. The fixer of <math>S</math> is the set of all <math>a\in M</math> fow which <math>a(x) = x</math> for all <math>x \in A</math>. If <math>S</math> contains a single element <math>x</math>, we sometimes call this the fixer or [[stabilizer]] of <math>x</math>, by abuse of language. |
Latest revision as of 18:47, 9 September 2008
A fixer is part of a monoid (or group) acting on a set.
Specifically, let be a monoid acting on
; let
be a subset of
. The fixer of
is the set of all
fow which
for all
. If
contains a single element
, we sometimes call this the fixer or stabilizer of
, by abuse of language.
Evidently, the fixer of is a submonoid of
(and of the strict stabilizer of
). Also, if
is an invertible element of the fixer of
, then
is evidently an element of the fixer of
.
It follows that if is a group
, then the fixer
of
is a subgroup of
. In fact, it is a normal subgroup of the strict stabilizer of
. Indeed, it is the kernel of the canonical homomorphism from the strict stabilizer of
to $\mathfrak{S}_A}$ (Error compiling LaTeX. Unknown error_msg), the group of permutations on
.
Note, however, that need not be a normal subgroup of
. For example, if
is
, the group of permutations acting on a set of size three, then the fixer of any element of the set is isomorphic to
and is not a normal subgroup of
.