Can a group be recovered from its order and automorphism?

Question

Does there exists two non-isomorphic finite groups $G$ and $H$ of same cardinality with $Aut(G) \cong Aut(H)$ ?

I know that the non-cyclic group of order $4$ i.e. $\mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$ and the non-abelian group of order $6$ i.e $S_{3}$ have isomorphic automorphism group. But I am interested in groups with same order? I don't know whether such example exists or not?

score 12 · Answer 1 · answered Jun 15 '19 at 04:35

12

Consider $G=Z_2\times Z_4$ and $H=D_8$.
Then $G\not\cong H$ but Aut$(G)\cong$ Aut$(H)\cong D_8$

answered Jun 15 '19 at 04:35

Wang Kah Lun

10,240

Thanks! for this example, Can it possible two find non abelian groups with the same property above? – Sunny Jun 15 '19 at 04:43
6

Yes, generalized quaternion and dihedral groups of order $2^n$ for $n>3$ have isomorphic automorphism groups. – Derek Holt Jun 15 '19 at 07:27
oh really! thanks – Sunny Jun 15 '19 at 14:06

Can a group be recovered from its order and automorphism?

1 Answers1