# On order automorphisms group of a finite group

Let $G$ be finite group and $Aut(G)$ denote the automorphisms group of $G$.

1) If $\mid G\mid>2$ is even, then do $\mid Aut(G)\mid$ is even?

2) Do we can determine structure all finite groups such that the order of the automorphisms group is even?

• If $C_2$ is the cyclic group of order two then $\operatorname{Aut}(C_2)$ is trivial, and so has odd order. – user1729 Apr 10 '13 at 15:59
• Ok. I forgot to write that $\mid G\mid >2$. I must edit it. – maryam Apr 10 '13 at 16:02
• Do we need the condition $|G|$ is even? (Interestingly, GAP says we do. Without this condition, the smallest non-trivial counterexample is of order $3^6=729$, and there's many of this order.) – Douglas S. Stones Apr 10 '13 at 16:03
• More seriously, $G/Z(G)$ is a subgroup of $\operatorname{Aut}(G)$. Thus, if $G/Z(G)$ is even then so is $\operatorname{Aut}(G)$. For example, if $G$ is a non-abelian group of order $2^n$ then $\operatorname{Aut}(G)$ is even. – user1729 Apr 10 '13 at 16:03
• @DouglasS.Stones: No. Take $C_3\times C_3$. – user1729 Apr 10 '13 at 16:04

a group $H$ of order $3 \times 19 \times 7^{12}$ is constructed, which has trivial centre and is isomorphic to its own automorphism group. So $H \times C_2$ has even order but with odd order automorphism group.
Note that it $|G|$ is even and $|G/Z(G)|$ is odd, then $G$ is a direct product of a group of odd order and an abelian 2-group $T$, and if $|{\rm Aut}(G)|$ is odd then we must have $|T|=2$. So all examples must be similar this one (although the odd order direct factor would not necessarily have to be complete).
• The smallest examples of groups with nontrivial automorphism groups of odd order are of order $3^6$. For example, $G = \operatorname{SmallGroup}(3^6, 90)$ has an automorphism group of order $3^7$. So by your remark, $C_2 \times G$ is a smallest possible example of even order. Reference: D. MacHale and R. Sheehy, Finite groups with odd order automorphism groups, Math. Proc. R. Ir. Acad. 95A (1995), no. 2, 113–116. JSTOR – Mikko Korhonen Apr 10 '13 at 17:19