group actions and homomorphisms
Notes on group actions and homomorphisms
Now let be a group homomorphism, and let satisfy
for all and
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
and it follows
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
and therefore is a monomorphism.
For the trivial operation of on given by the stabilizer subgroup is for all , and thus
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
is faithful but not free.
|Title||group actions and homomorphisms|
|Date of creation||2013-03-22 13:18:48|
|Last modified on||2013-03-22 13:18:48|
|Last modified by||CWoo (3771)|