# Automorphisms and Permutations

• Nov 25th 2011, 07:11 PM
Bernhard
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 $\displaystyle S_G$"

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
• Nov 25th 2011, 07:48 PM
Drexel28
Re: Automorphisms and Permutations
Quote:

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 $\displaystyle S_G$"

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

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 $\displaystyle \sigma:G\to G$ fixes $\displaystyle 1$. This automatically knocks out, best case scenario, an $\displaystyle |G|^{\text{th}}$ of the permutations.
• Nov 25th 2011, 07:50 PM
Bernhard
Re: Automorphisms and Permutations
Thanks

Yes, getting the idea

Peter