When is Inn(G) abelian or cyclic.

Asked Oct 21 '16 at 22:43

Active Sep 19 '19 at 23:03

Viewed 791 times

I proved Inn(G) cyclic implies G abelian.

and if G finite group, then Aut(G) cyclic iff G cyclic and order G is either 1,2, 4 or $p^k$ or 2$p^k$.

Is there chacterisation for(given G is finite group):

Question 1: When Inn(G) is cyclic?

Question 2: When Inn(G) is abelian?

Question 3: When Aut(G) is abelian?

edited Sep 19 '19 at 23:03

Shaun

asked Oct 21 '16 at 22:43

Sushil

Question 1: see here. Question 2: see here. Question 3: see here. – Dietrich Burde Oct 21 '16 at 22:48
@DietrichBurde thanks but for question 2 atleast you only mentioned sufficinet condition for Inn(G) abelian. Is there a necessary condition also – Sushil Oct 22 '16 at 00:04
1

$\textrm{Inn}(G)$ is abelian iff $G/Z(G)$ is abelian iff $G$ is at most $2$-step nilpotent. – Keith Kearnes Oct 22 '16 at 13:48
sorry not familiar with term 2-step nilpotent. Can you explain or give soem reference @KeithKearnes – Sushil Oct 22 '16 at 19:36
1

https://en.wikipedia.org/wiki/Nilpotent_group – Keith Kearnes Oct 23 '16 at 08:47

0 Answers0