group actions and homomorphisms
group actions and homomorphisms
Notes on group actions and homomorphisms
Let be a group, a non-empty set and the symmetric group
of ,
i.e. the group of all bijective maps on . may denote a left
group
action of on .
- 1.
For each and we define
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Since for each
,
is the inverse of . so is bijective and thus element of
. We define for all . This
mapping is a group homomorphism: Let . Then
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for all implies . — The same is obviously true for a right
group action.
- 2.
Now let be a group homomorphism, and let
satisfy
- (a)
for all and
- (b)
,
so is a group action induced by .
Characterization of group actions
Let be a group acting on a set .
Using the same notation as above, we have for each
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(1) |
and it follows
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Let act transitively on . Then for any , is the
orbit
of . As shown in “conjugate stabilizer subgroups’, all stabilizer
subgroups
of elements are conjugate subgroups to in . From
the above it follows that
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For a faithful operation of the condition is equivalent to
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and therefore is a monomorphism.
For the trivial operation of on given by
the stabilizer subgroup is for all , and thus
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If the operation of on is free, then
, thus the kernel of is
–like for a faithful operation. But:
Let and . Then the operation of on
given by
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is faithful but not free.
Mathematics Subject Classification
20A05
Axiomatics and elementary properties