you can check this link there are several examples
Characteristic subgroup - Wikipedia, the free encyclopedia
Can anyone please help with the following problem from Dummit and Foote:
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Prove that characteristic subgroups are normal. Give an example of a normal subgroup that is not characteristic.
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Peter
you can check this link there are several examples
Characteristic subgroup - Wikipedia, the free encyclopedia
if H is a characteristic subgroup of G, then for every φ in Aut(G), φ(H) = H. in particular, if φ is an inner automorphism, φ(x) = gxg^-1,
then φ(H) = gHg^-1 = H. since this is true of EVERY inner automorphism (for a characteristic subgroup), H is normal.
an easy example of a normal subgroup which is NOT characteristic is the subgroup H = {1,r^2,s,r^2s} of D8. it's normal because it is of index two.
but if φ is the automorphism that sends r-->r and s-->rs, then φ(H) = {1,r^2,rs,r^3s} ≠ H.
Yes, agree.
However the definition of characteristic subgroup in Dummit and Foote (page 135) is as follows:
Defintion. A sugroup H of a group G is called characteristic in G , denoted H char G if *every* automorphism of G maps H to itself i.e. (H) = H for all Aut(G).
My problem or worry with Deveno's proof is that he seems to prove that (H) = H for all inner automorphisms but ... we need to prove the case for all automorphisms ...
Am I right ... or perhaps missing something ...
Can someone please clarify this matter
Peter
the image of any normal subgroup in any inner automorphism is the normal subgroup
let
normal in G and inner automorphism
for any h in H
1-1 onto so will map H to H
for the characteristic subgroup that holds for any auto should not be inner auto
i.e for normal subgroup you may find an auto which will not map the normal subgroup to itself
I hope it is clear
Thanks I have looked at it again and think my logic was muddled
Deveno showed that if we take an inner automorphism (a particular type of automorphism) then the charactersitic subgrup was normal - so that is the end of the story I think - he does not have to prove anything more
Peter
another way of saying this is:
normal = fixed by inner automorphisms
characteristic = fixed by inner and outer automorphisms
which makes it clear that all characteristic subgroups are normal, but not necessarily the other way 'round.
Said well, support you.
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Schaufensterpuppen