Is there non-trivial examples for groups $G$ satisfying $G = \mathrm{Aut}(G)$

Question

As the question says. Are there non-trivial groups isomorphic to their automorphisms group?

(By the way i am İbrahim İpek but i lost the password of my account and also mail :/ Is there anything anyone can do?)

I think you can ask the moderators to merge your old and new accounts. — Angina Seng, Dec 11 '19 at 20:51

score 6 · Accepted Answer · answered Dec 11 '19 at 20:51

6

And for an example where $Z(G)\neq \{e\}$, so that the automorphism group is not just the inner automorphism group, take the Dihedral group of order $8$.

answered Dec 11 '19 at 20:51

Arturo Magidin

398,050

score 4 · Answer 2 · answered Dec 11 '19 at 21:00

4

If $G$ is a complete group, that is, with trivial center and no outer automorphisms, then $$G\simeq \text{Aut}(G)$$ as every automorphism is the conjugation by some element, and the map $g\mapsto g\cdot g^{-1}$ has a trivial kernel.

answered Dec 11 '19 at 21:00

Arnaud Mortier

27,276

Con · Answer 3 · 2019-12-11T20:54:28.550

3

Yes, for example we have $\text{Aut}(S_n) \cong S_n$ if $n \neq 2,6$.

edited Dec 11 '19 at 20:54

answered Dec 11 '19 at 20:48

Con

9,000

Is there non-trivial examples for groups $G$ satisfying $G = \mathrm{Aut}(G)$

3 Answers3