Characteristic subgroups are normal

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

Re: Characteristic subgroups are normal

Re: Characteristic subgroups are normal

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.

Re: Characteristic subgroups are normal

Thanks for the help

Still puzzling over this however.

We have to show that every characteristic group is normal - so surely we have to consider automorphisms which are not inner automorphisms?

Can you help?

Peter

Re: Characteristic subgroups are normal

inner automorphisms is a type of the automorphisms

Re: Characteristic subgroups are normal

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. $\displaystyle \phi $(H) = H __ for all __$\displaystyle \phi$$\displaystyle \in$ Aut(G).

My problem or worry with Deveno's proof is that he seems to prove that $\displaystyle \phi$(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

Re: Characteristic subgroups are normal

the image of any normal subgroup in any inner automorphism is the normal subgroup

let

$\displaystyle H$ normal in G and $\displaystyle \phi $ inner automorphism

$\displaystyle \phi : G \rightarrow G $

$\displaystyle \phi (x) = g x g^{-1} $

for any h in H

$\displaystyle \phi (h) = ghg^{-1} \in H $ 1-1 onto so $\displaystyle \phi $ 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

Re: Characteristic subgroups are normal

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

Re: Characteristic subgroups are normal

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.

Re: Characteristic subgroups are normal

Thanks Deveno

Yes, that makes it very clear.

Peter

Re: Characteristic subgroups are normal

Quote:

Originally Posted by

**Bernhard** 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

Said well, support you. (Hi)

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