Talk:Outer automorphism group

Simple groups of Lie type

What is the precise structures of Outer automorphism groups of Lie type simple groups? —Preceding unsigned comment added by 219.138.209.67 (talk • contribs)

Integers

Tha table says that Out(Z) = Z_2, but all the definitions lead to Out(Z) = Z. In fact, Inn(Z) = 0, since Z is abelian. Which is the reference used for the table? Espigaymostaza (talk) 14:40, 11 June 2008 (UTC)

Aut(Z) = Units(Z) = {+1,-1} is cyclic of order two. Hence Out(Z) = Aut(Z)/Inn(Z) =~ Aut(Z) is cyclic of order two. The table is likely pulled from several references. The cleanup tag "fact" has been added to indicate someone should go and record which references. JackSchmidt (talk) 15:03, 11 June 2008 (UTC)

Outer Automorphisms are not themselves automorphisms?

The page says that because we have a quotient space, which is not necessarily a subgroup (correct) that the outer automorphisms are not automorphisms. That seems contradictory to the definition which is that any automorphism which is not inner is outer. I think this is just a mistake, but I'm not an expert, so could someone who knows please edit it? — Preceding unsigned comment added by 152.1.205.201 (talk) 15:19, 24 April 2012 (UTC)

According to chapter 7 of the book An introduction to the theory of groups by Rotman, an outer automorphism is an automorphism that is not inner. So the elements of Out(G) should not be called outer automorphisms. --Stomatapoll (talk) 19:36, 20 June 2013 (UTC)

[Droste–Giraudet–Göbel 2001]: Every group is the outer automorphism group of a simple group.

Seen on twitter today. Seems it should be noted here. Seems remarkable. Seems to be constructive, too: "How hard is it to find the simple group whose outer automorphism group is Z/n, or Sₙ? Easy... Let m > 2. Then PSp(2m,2^n) has outer automorphism group cyclic of order n. ...the automorphism is induced by the Frobenius automorphism x -> x^2 on GF(2^n)." Unclear to me if its always constructive... 84.15.181.197 (talk) 07:21, 17 August 2022 (UTC)