7
$\begingroup$

What is the automorphism group of direct sum of dihedral group of order $8$, $D_4$?

For example, $\mathrm{Aut}(D_4)$ is isomorphic to $D_4$. How about $\mathrm{Aut}(D_4\times D_4)$, $\mathrm{Aut}(D_4\times D_4\times D_4)$, and $\mathrm{Aut}(D_4\times D_4 \times D_4 \times D_4)$?

$\endgroup$
2
  • $\begingroup$ Is $D_4$ the dihedral group of order $8$? $\endgroup$
    – LSpice
    Aug 26, 2018 at 23:47
  • $\begingroup$ @LSpice Yes it is. $\endgroup$
    – Sirui Lu
    Aug 26, 2018 at 23:47

2 Answers 2

11
$\begingroup$

The following papers are relevant:

[1] J. N. S. Bidwell, M. J. Curran, and D. J. McCaughan, Automorphisms of direct products of finite groups, Arch. Math. 86, 481 – 489 (2006).

[2] J. N. S. Bidwell, Automorphisms of direct products of finite groups II, Arch. Math. 91, 111–121 (2008).

For your question you want to look at [2]. This paper describes the automorphism group of $G = H^n = H \times \cdots \times H$ where $H$ is an indecomposable non-abelian group. In this case $\operatorname{Aut}(G)$ has a normal subgroup $\mathscr{A}$ isomorphic to the group formed by the matrices $$\left\{ \begin{pmatrix} \alpha_{11} & \cdots & \alpha_{1n} \\ \vdots & \ddots & \vdots \\ \alpha_{n1} & \cdots & \alpha_{nn}\end{pmatrix} : \begin{align}\alpha_{ii} &\in \operatorname{Aut}(H) \text{ for all } 1 \leq i \leq n \\ \alpha_{ij} &\in \operatorname{Hom}(H, Z(H)) \text{ for all i $\neq$ j} \end{align}\right\}.$$

(The group operation is matrix multiplication, with multiplication defined by composition and addition defined by $(\alpha+\beta)(x) = \alpha(x)\beta(x)$.)

Theorem 3.1 of [2] states that $\operatorname{Aut}(G) = \mathscr{A} \rtimes S_n$, where $S_n$ is the symmetric group acting on $G$ by permuting the direct factors. Thus $|\operatorname{Aut}(G)| = |\operatorname{Aut}(H)|^n |\operatorname{Hom}(H, Z(H))|^{n^2-n} n!$

In your case $\operatorname{Aut}(H) \cong D_4$ and $\operatorname{Hom}(H, Z(H)) \cong C_2 \times C_2$, so $|\operatorname{Aut}(G)| = 2^{2n^2+n} n!$.

$\endgroup$
2
  • $\begingroup$ Thanks for your answer! Can we get things beyond order, like generators of G? $\endgroup$
    – Sirui Lu
    Aug 27, 2018 at 2:39
  • 1
    $\begingroup$ @SiruiLu: Not sure what you mean. If you know $\operatorname{Aut}(H)$ and $\operatorname{Hom}(H, Z(H))$, you have an explicit description of all the automorphisms of $G$, not just the number of them. $\endgroup$ Aug 27, 2018 at 2:53
2
$\begingroup$

Mikko Korhonen has already given a good answer. -- But as you asked for explicit generators for the automorphism groups -- you can obtain such by GAP as follows (you see that 4 generators suffice):

gap> D4 := Group((1,2,3,4),(1,3));
Group([ (1,2,3,4), (1,3) ])
gap> A1 := AutomorphismGroup(D4);                  
<group of size 8 with 3 generators>
gap> SmallGeneratingSet(A1);
[ [ (2,4), (1,4)(2,3) ] -> [ (1,2)(3,4), (2,4) ], 
  [ (2,4), (1,4)(2,3) ] -> [ (2,4), (1,2)(3,4) ] ]
gap> A2 := AutomorphismGroup(DirectProduct(D4,D4));
<group of size 2048 with 11 generators>
gap> SmallGeneratingSet(A2);                       
[ [ (1,3)(2,4)(6,8), (2,4)(5,7)(6,8), (1,2)(3,4), (1,4)(2,3)(5,7)(6,8), 
      (1,3)(2,4)(5,8)(6,7) ] -> [ (1,3)(2,4)(6,8), (2,4)(5,7)(6,8), 
      (1,2)(3,4)(5,7)(6,8), (1,4)(2,3), (1,3)(2,4)(5,8)(6,7) ], 
  [ (1,3)(2,4)(6,8), (2,4)(5,7)(6,8), (1,2)(3,4), (1,4)(2,3)(5,7)(6,8), 
      (1,3)(2,4)(5,8)(6,7) ] -> [ (1,3)(2,4)(6,8), (2,4)(5,7)(6,8), 
      (1,4)(2,3), (1,2)(3,4)(5,7)(6,8), (1,3)(2,4)(5,8)(6,7) ], 
  [ (1,3)(2,4)(6,8), (2,4)(5,7)(6,8), (1,2)(3,4), (1,4)(2,3)(5,7)(6,8), 
      (1,3)(2,4)(5,8)(6,7) ] -> [ (1,2)(3,4)(5,7)(6,8), (1,3)(2,4)(6,8), 
      (5,6)(7,8), (1,3)(2,4)(5,8)(6,7), (2,4)(5,7)(6,8) ], 
  [ (1,3)(2,4)(6,8), (2,4)(5,7)(6,8), (1,2)(3,4), (1,4)(2,3)(5,7)(6,8), 
      (1,3)(2,4)(5,8)(6,7) ] -> [ (1,3)(2,4)(5,6)(7,8), (2,4)(5,7)(6,8), 
      (1,2)(3,4), (1,4)(2,3)(5,7)(6,8), (1,3)(2,4)(6,8) ] ]
gap> A3 := AutomorphismGroup(DirectProduct(D4,D4,D4));
<group of size 12582912 with 23 generators>
gap> SmallGeneratingSet(A3);                          
[ [ (1,2)(3,4)(6,8)(9,11), (1,2,3,4)(5,6)(7,8)(9,10)(11,12), 
      (1,2)(3,4)(5,6)(7,8)(10,12), (1,3)(5,6)(7,8)(9,10)(11,12), 
      (1,4)(2,3)(6,8)(9,12)(10,11), (1,2)(3,4)(5,8)(6,7)(9,10,11,12) ] -> 
    [ (2,4)(5,6)(7,8)(10,12), (1,4)(2,3)(5,6,7,8)(9,10)(11,12), 
      (1,2)(3,4)(5,6)(7,8)(10,12), (1,4)(2,3)(6,8)(9,12)(10,11), 
      (1,3)(5,6)(7,8)(9,10)(11,12), (1,4)(2,3)(5,6)(7,8)(9,12,11,10) ], 
  [ (1,2)(3,4)(6,8)(9,11), (1,2,3,4)(5,6)(7,8)(9,10)(11,12), 
      (1,2)(3,4)(5,6)(7,8)(10,12), (1,3)(5,6)(7,8)(9,10)(11,12), 
      (1,4)(2,3)(6,8)(9,12)(10,11), (1,2)(3,4)(5,8)(6,7)(9,10,11,12) ] -> 
    [ (1,3)(5,8)(6,7)(9,10)(11,12), (1,2,3,4)(6,8)(9,11), 
      (2,4)(6,8)(9,10)(11,12), (1,2)(3,4)(5,7)(10,12), (2,4)(5,8)(6,7)(9,11), 
      (2,4)(5,7)(9,10,11,12) ], 
  [ (1,2)(3,4)(6,8)(9,11), (1,2,3,4)(5,6)(7,8)(9,10)(11,12), 
      (1,2)(3,4)(5,6)(7,8)(10,12), (1,3)(5,6)(7,8)(9,10)(11,12), 
      (1,4)(2,3)(6,8)(9,12)(10,11), (1,2)(3,4)(5,8)(6,7)(9,10,11,12) ] -> 
    [ (2,4)(5,8)(6,7)(9,11), (1,4)(2,3)(5,6,7,8)(9,10)(11,12), 
      (2,4)(5,8)(6,7)(9,10)(11,12), (1,4)(2,3)(6,8)(9,10)(11,12), 
      (1,2)(3,4)(5,8)(6,7)(10,12), (1,2,3,4)(5,6)(7,8)(9,12)(10,11) ], 
  [ (1,2)(3,4)(6,8)(9,11), (1,2,3,4)(5,6)(7,8)(9,10)(11,12), 
      (1,2)(3,4)(5,6)(7,8)(10,12), (1,3)(5,6)(7,8)(9,10)(11,12), 
      (1,4)(2,3)(6,8)(9,12)(10,11), (1,2)(3,4)(5,8)(6,7)(9,10,11,12) ] -> 
    [ (1,3)(5,7)(9,10)(11,12), (1,4,3,2)(5,8)(6,7)(10,12), 
      (1,3)(5,6)(7,8)(9,10)(11,12), (1,2)(3,4)(5,6)(7,8)(9,11), 
      (1,3)(5,7)(9,11), (1,3)(5,6)(7,8)(9,12,11,10) ] ]
gap> A4 := AutomorphismGroup(DirectProduct(D4,D4,D4,D4));
<group of size 1649267441664 with 40 generators>
gap> SmallGeneratingSet(A4);                             
[ [ (2,4)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15), 
      (5,6)(7,8)(9,11)(13,16)(14,15), (1,2)(3,4)(6,8)(9,10)(11,12)(13,15)(14,
        16), (1,3)(5,8)(6,7)(9,11)(10,12)(13,15), (2,4)(6,8)(10,12), 
      (1,3)(2,4)(5,6)(7,8)(9,12)(10,11)(14,16), (2,4)(5,8)(6,7)(9,10)(11,12), 
      (1,3)(5,7)(6,8)(10,12)(13,14)(15,16) ] -> 
    [ (2,4)(5,6)(7,8)(9,12)(10,11)(13,15)(14,16), 
      (5,6)(7,8)(9,12)(10,11)(13,15), (1,2)(3,4)(6,8)(9,11)(10,12)(13,14)(15,
        16), (1,3)(5,8)(6,7)(9,11)(13,15)(14,16), (2,4)(6,8)(14,16), 
      (1,3)(2,4)(5,6)(7,8)(10,12)(13,16)(14,15), 
      (2,4)(5,8)(6,7)(13,14)(15,16), (1,3)(5,7)(6,8)(9,10)(11,12)(14,16) ], 
  [ (2,4)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15), 
      (5,6)(7,8)(9,11)(13,16)(14,15), (1,2)(3,4)(6,8)(9,10)(11,12)(13,15)(14,
        16), (1,3)(5,8)(6,7)(9,11)(10,12)(13,15), (2,4)(6,8)(10,12), 
      (1,3)(2,4)(5,6)(7,8)(9,12)(10,11)(14,16), (2,4)(5,8)(6,7)(9,10)(11,12), 
      (1,3)(5,7)(6,8)(10,12)(13,14)(15,16) ] -> 
    [ (1,3)(5,7)(6,8)(9,10)(11,12)(13,14)(15,16), (1,3)(5,7)(9,10)(11,12), 
      (1,3)(2,4)(5,6)(7,8)(10,12)(14,16), 
      (1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15), 
      (6,8)(10,12)(13,14)(15,16), (1,4)(2,3)(5,8)(6,7)(9,10)(11,12)(13,15)(14,
        16), (5,6)(7,8)(9,12)(10,11)(13,14)(15,16), 
      (2,4)(6,8)(9,11)(10,12)(13,16)(14,15) ], 
  [ (2,4)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15), 
      (5,6)(7,8)(9,11)(13,16)(14,15), (1,2)(3,4)(6,8)(9,10)(11,12)(13,15)(14,
        16), (1,3)(5,8)(6,7)(9,11)(10,12)(13,15), (2,4)(6,8)(10,12), 
      (1,3)(2,4)(5,6)(7,8)(9,12)(10,11)(14,16), (2,4)(5,8)(6,7)(9,10)(11,12), 
      (1,3)(5,7)(6,8)(10,12)(13,14)(15,16) ] -> 
    [ (1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,14)(15,16), 
      (5,6)(7,8)(9,11)(13,14)(15,16), (2,4)(6,8)(9,10)(11,12)(13,15)(14,16), 
      (1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(14,16), (1,4)(2,3)(6,8)(10,12), 
      (1,3)(2,4)(5,6)(7,8)(9,12)(10,11)(13,15), 
      (1,4)(2,3)(5,8)(6,7)(9,10)(11,12), 
      (1,2)(3,4)(5,7)(6,8)(10,12)(13,16)(14,15) ], 
  [ (2,4)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15), 
      (5,6)(7,8)(9,11)(13,16)(14,15), (1,2)(3,4)(6,8)(9,10)(11,12)(13,15)(14,
        16), (1,3)(5,8)(6,7)(9,11)(10,12)(13,15), (2,4)(6,8)(10,12), 
      (1,3)(2,4)(5,6)(7,8)(9,12)(10,11)(14,16), (2,4)(5,8)(6,7)(9,10)(11,12), 
      (1,3)(5,7)(6,8)(10,12)(13,14)(15,16) ] -> 
    [ (1,3)(2,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16), 
      (1,2)(3,4)(5,8)(6,7)(9,10)(11,12), (1,3)(5,7)(9,11)(10,12)(14,16), 
      (1,3)(2,4)(5,8)(6,7)(9,11)(13,16)(14,15), 
      (1,2)(3,4)(5,7)(9,11)(10,12)(13,16)(14,15), 
      (1,3)(5,6)(7,8)(10,12)(13,15)(14,16), (2,4)(5,8)(6,7)(13,14)(15,16), 
      (1,4)(2,3)(9,12)(10,11)(13,16)(14,15) ] ]
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.