2

Let $G$ be a group such that we Know that there exists some group $H$ such that $$G\cong Inn(H),$$ where Inn(G) is Inner automorphisms Group of $H$.

Now How by GAP we can find $H$?

Thank you

Olexandr Konovalov
  • 7,186
  • 2
  • 35
  • 73
A.G
  • 839
  • 2
    If $G\cong Inn(H)$ then also, $G\cong Inn(H\times A)$ for any abelian group $A$. Hence $H$ is not uniquely determined by $G$. The only possible way could be checking some properties of $G$, and some trial-and-error method. – p Groups Mar 27 '16 at 09:25
  • @pGroups: Yes there exists many groups $H$, such that $G\cong Inn(H)$. I want only one group $H$ such that $G\cong Inn(H).$ – A.G Mar 27 '16 at 12:57
  • The question in the title doesn't seem to match the question in the body. – Daniel McLaury Mar 27 '16 at 20:01

1 Answers1

2

It is possible to find an example of such group using the GAP Small Groups Library.

First one should find out how to check this property for a given group. Typing ??Inner in GAP, you should be able to find InnerAutomorphismsAutomorphismGroup and AutomorphismGroup and then proceed like suggested in the discussion here. I suggest to check first that the order of $Inn(G)$ is equal to the order of $G$ - clearly, if not, then they are non-isomorphic.

As soon as you can check this property for a single group, you can start to search systematically through the GAP small groups library - see e.g. an example from my GAP Software Carpentry lesson here.

Community
  • 1
Olexandr Konovalov
  • 7,186
  • 2
  • 35
  • 73