Automorphisms and Permutations

I am reading Dummit and Foote Section 4.4 on Automorphisms.

In the second papragraph of the secion Dummit and Foote write:

"Notice that automorphisms of a group G are, in particular, permutations of the set G so Aut(G) is a subgroup of "

Intuitively it seems to me, at first reflection, at least, that any permututation of G would be an automorphism of G, but based on D&F's statement this is not the case

Can anyone with more experience of algebra explain at a broad intuitive level why this is not the case?

Peter

Quote:

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Originally Posted by Bernhard

I am reading Dummit and Foote Section 4.4 on Automorphisms.

In the second papragraph of the secion Dummit and Foote write:

"Notice that automorphisms of a group G are, in particular, permutations of the set G so Aut(G) is a subgroup of "

Intuitively it seems to me, at first reflection, at least, that any permututation of G would be an automorphism of G, but based on D&F's statement this is not the case

Can anyone with more experience of algebra explain at a broad intuitive level why this is not the case?

Peter

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I don't know how much intuition you want. Automorphisms are permutations that must satisfy EXTRA, NON-trivial properties. The extra tells us that every automorphism is, first and foremost, a permutation and the NON tells us that not every permutation satisfies this extra condition. For example, an automorphism fixes . This automatically knocks out, best case scenario, an of the permutations.

Thanks

Yes, getting the idea

Peter