What are the automorphism groups of direct products of dihedral group D4
$begingroup$
What is the automorphism group of direct sum of dihedral group of order $8$, $D_4$?
For example, $mathrmAut(D_4)$ is isomorphic to $D_4$. How about $mathrmAut(D_4times D_4)$, $mathrmAut(D_4times D_4times D_4)$, and $mathrmAut(D_4times D_4 times D_4 times D_4)$?
gr.group-theory finite-groups automorphism-groups
edited Aug 27 '18 at 3:08
Martin Sleziak
2,98532128
asked Aug 26 '18 at 22:43
Sirui LuSirui Lu
412
$endgroup$
add a comment |
$begingroup$
What is the automorphism group of direct sum of dihedral group of order $8$, $D_4$?
For example, $mathrmAut(D_4)$ is isomorphic to $D_4$. How about $mathrmAut(D_4times D_4)$, $mathrmAut(D_4times D_4times D_4)$, and $mathrmAut(D_4times D_4 times D_4 times D_4)$?
gr.group-theory finite-groups automorphism-groups
edited Aug 27 '18 at 3:08
Martin Sleziak
2,98532128
asked Aug 26 '18 at 22:43
Sirui LuSirui Lu
412
$endgroup$
$begingroup$
Is $D_4$ the dihedral group of order $8$?
$endgroup$
– LSpice
Aug 26 '18 at 23:47
$begingroup$
@LSpice Yes it is.
$endgroup$
– Sirui Lu
Aug 26 '18 at 23:47
add a comment |
$begingroup$
What is the automorphism group of direct sum of dihedral group of order $8$, $D_4$?
For example, $mathrmAut(D_4)$ is isomorphic to $D_4$. How about $mathrmAut(D_4times D_4)$, $mathrmAut(D_4times D_4times D_4)$, and $mathrmAut(D_4times D_4 times D_4 times D_4)$?
gr.group-theory finite-groups automorphism-groups
edited Aug 27 '18 at 3:08
Martin Sleziak
2,98532128
asked Aug 26 '18 at 22:43
Sirui LuSirui Lu
412
$endgroup$
What is the automorphism group of direct sum of dihedral group of order $8$, $D_4$?
For example, $mathrmAut(D_4)$ is isomorphic to $D_4$. How about $mathrmAut(D_4times D_4)$, $mathrmAut(D_4times D_4times D_4)$, and $mathrmAut(D_4times D_4 times D_4 times D_4)$?
gr.group-theory finite-groups automorphism-groups
gr.group-theory finite-groups automorphism-groups
edited Aug 27 '18 at 3:08
Martin Sleziak
2,98532128
asked Aug 26 '18 at 22:43
Sirui LuSirui Lu
412
edited Aug 27 '18 at 3:08
Martin Sleziak
2,98532128
asked Aug 26 '18 at 22:43
Sirui LuSirui Lu
412
edited Aug 27 '18 at 3:08
Martin Sleziak
2,98532128
edited Aug 27 '18 at 3:08
Martin Sleziak
2,98532128
edited Aug 27 '18 at 3:08
Martin Sleziak
2,98532128
2,98532128
asked Aug 26 '18 at 22:43
Sirui LuSirui Lu
412
asked Aug 26 '18 at 22:43
Sirui LuSirui Lu
412
asked Aug 26 '18 at 22:43
Sirui LuSirui Lu
412
412
$begingroup$
Is $D_4$ the dihedral group of order $8$?
$endgroup$
– LSpice
Aug 26 '18 at 23:47
$begingroup$
@LSpice Yes it is.
$endgroup$
– Sirui Lu
Aug 26 '18 at 23:47
add a comment |
$begingroup$
Is $D_4$ the dihedral group of order $8$?
$endgroup$
– LSpice
Aug 26 '18 at 23:47
$begingroup$
@LSpice Yes it is.
$endgroup$
– Sirui Lu
Aug 26 '18 at 23:47
$begingroup$
Is $D_4$ the dihedral group of order $8$?
$endgroup$
– LSpice
Aug 26 '18 at 23:47
$begingroup$
Is $D_4$ the dihedral group of order $8$?
$endgroup$
– LSpice
Aug 26 '18 at 23:47
$begingroup$
@LSpice Yes it is.
$endgroup$
– Sirui Lu
Aug 26 '18 at 23:47
$begingroup$
@LSpice Yes it is.
$endgroup$
– Sirui Lu
Aug 26 '18 at 23:47
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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 $operatornameAut(G)$ has a normal subgroup $mathscrA$ isomorphic to the group formed by the matrices $$left beginpmatrix alpha_11 & cdots & alpha_1n \ vdots & ddots & vdots \ alpha_n1 & cdots & alpha_nnendpmatrix : beginalignalpha_ii &in operatornameAut(H) text for all 1 leq i leq n \ alpha_ij &in operatornameHom(H, Z(H)) text for all i $neq$ j endalignright.$$
(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 $operatornameAut(G) = mathscrA rtimes S_n$, where $S_n$ is the symmetric group acting on $G$ by permuting the direct factors. Thus $|operatornameAut(G)| = |operatornameAut(H)|^n |operatornameHom(H, Z(H))|^n^2-n n!$
In your case $operatornameAut(H) cong D_4$ and $operatornameHom(H, Z(H)) cong C_2 times C_2$, so $|operatornameAut(G)| = 2^2n^2+n n!$.
edited Aug 27 '18 at 3:18
Martin Sleziak
2,98532128
answered Aug 27 '18 at 2:36
Mikko KorhonenMikko Korhonen
1,070914
$endgroup$
$begingroup$
Thanks for your answer! Can we get things beyond order, like generators of G?
$endgroup$
– Sirui Lu
Aug 27 '18 at 2:39
1
$begingroup$
@SiruiLu: Not sure what you mean. If you know $operatornameAut(H)$ and $operatornameHom(H, Z(H))$, you have an explicit description of all the automorphisms of $G$, not just the number of them.
$endgroup$
– Mikko Korhonen
Aug 27 '18 at 2:53
add a comment |
$begingroup$
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) ] ]
answered Sep 15 '18 at 15:08
Stefan KohlStefan Kohl
12.5k956112
$endgroup$
add a comment |
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2 Answers
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active
oldest
votes
2 Answers
2
active
oldest
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active
oldest
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active
oldest
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$begingroup$
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 $operatornameAut(G)$ has a normal subgroup $mathscrA$ isomorphic to the group formed by the matrices $$left beginpmatrix alpha_11 & cdots & alpha_1n \ vdots & ddots & vdots \ alpha_n1 & cdots & alpha_nnendpmatrix : beginalignalpha_ii &in operatornameAut(H) text for all 1 leq i leq n \ alpha_ij &in operatornameHom(H, Z(H)) text for all i $neq$ j endalignright.$$
(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 $operatornameAut(G) = mathscrA rtimes S_n$, where $S_n$ is the symmetric group acting on $G$ by permuting the direct factors. Thus $|operatornameAut(G)| = |operatornameAut(H)|^n |operatornameHom(H, Z(H))|^n^2-n n!$
In your case $operatornameAut(H) cong D_4$ and $operatornameHom(H, Z(H)) cong C_2 times C_2$, so $|operatornameAut(G)| = 2^2n^2+n n!$.
edited Aug 27 '18 at 3:18
Martin Sleziak
2,98532128
answered Aug 27 '18 at 2:36
Mikko KorhonenMikko Korhonen
1,070914
$endgroup$
$begingroup$
Thanks for your answer! Can we get things beyond order, like generators of G?
$endgroup$
– Sirui Lu
Aug 27 '18 at 2:39
1
$begingroup$
@SiruiLu: Not sure what you mean. If you know $operatornameAut(H)$ and $operatornameHom(H, Z(H))$, you have an explicit description of all the automorphisms of $G$, not just the number of them.
$endgroup$
– Mikko Korhonen
Aug 27 '18 at 2:53
add a comment |
$begingroup$
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 $operatornameAut(G)$ has a normal subgroup $mathscrA$ isomorphic to the group formed by the matrices $$left beginpmatrix alpha_11 & cdots & alpha_1n \ vdots & ddots & vdots \ alpha_n1 & cdots & alpha_nnendpmatrix : beginalignalpha_ii &in operatornameAut(H) text for all 1 leq i leq n \ alpha_ij &in operatornameHom(H, Z(H)) text for all i $neq$ j endalignright.$$
(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 $operatornameAut(G) = mathscrA rtimes S_n$, where $S_n$ is the symmetric group acting on $G$ by permuting the direct factors. Thus $|operatornameAut(G)| = |operatornameAut(H)|^n |operatornameHom(H, Z(H))|^n^2-n n!$
In your case $operatornameAut(H) cong D_4$ and $operatornameHom(H, Z(H)) cong C_2 times C_2$, so $|operatornameAut(G)| = 2^2n^2+n n!$.
edited Aug 27 '18 at 3:18
Martin Sleziak
2,98532128
answered Aug 27 '18 at 2:36
Mikko KorhonenMikko Korhonen
1,070914
$endgroup$
$begingroup$
Thanks for your answer! Can we get things beyond order, like generators of G?
$endgroup$
– Sirui Lu
Aug 27 '18 at 2:39
1
$begingroup$
@SiruiLu: Not sure what you mean. If you know $operatornameAut(H)$ and $operatornameHom(H, Z(H))$, you have an explicit description of all the automorphisms of $G$, not just the number of them.
$endgroup$
– Mikko Korhonen
Aug 27 '18 at 2:53
add a comment |
$begingroup$
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 $operatornameAut(G)$ has a normal subgroup $mathscrA$ isomorphic to the group formed by the matrices $$left beginpmatrix alpha_11 & cdots & alpha_1n \ vdots & ddots & vdots \ alpha_n1 & cdots & alpha_nnendpmatrix : beginalignalpha_ii &in operatornameAut(H) text for all 1 leq i leq n \ alpha_ij &in operatornameHom(H, Z(H)) text for all i $neq$ j endalignright.$$
(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 $operatornameAut(G) = mathscrA rtimes S_n$, where $S_n$ is the symmetric group acting on $G$ by permuting the direct factors. Thus $|operatornameAut(G)| = |operatornameAut(H)|^n |operatornameHom(H, Z(H))|^n^2-n n!$
In your case $operatornameAut(H) cong D_4$ and $operatornameHom(H, Z(H)) cong C_2 times C_2$, so $|operatornameAut(G)| = 2^2n^2+n n!$.
edited Aug 27 '18 at 3:18
Martin Sleziak
2,98532128
answered Aug 27 '18 at 2:36
Mikko KorhonenMikko Korhonen
1,070914
$endgroup$
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 $operatornameAut(G)$ has a normal subgroup $mathscrA$ isomorphic to the group formed by the matrices $$left beginpmatrix alpha_11 & cdots & alpha_1n \ vdots & ddots & vdots \ alpha_n1 & cdots & alpha_nnendpmatrix : beginalignalpha_ii &in operatornameAut(H) text for all 1 leq i leq n \ alpha_ij &in operatornameHom(H, Z(H)) text for all i $neq$ j endalignright.$$
(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 $operatornameAut(G) = mathscrA rtimes S_n$, where $S_n$ is the symmetric group acting on $G$ by permuting the direct factors. Thus $|operatornameAut(G)| = |operatornameAut(H)|^n |operatornameHom(H, Z(H))|^n^2-n n!$
In your case $operatornameAut(H) cong D_4$ and $operatornameHom(H, Z(H)) cong C_2 times C_2$, so $|operatornameAut(G)| = 2^2n^2+n n!$.
edited Aug 27 '18 at 3:18
Martin Sleziak
2,98532128
answered Aug 27 '18 at 2:36
Mikko KorhonenMikko Korhonen
1,070914
edited Aug 27 '18 at 3:18
Martin Sleziak
2,98532128
edited Aug 27 '18 at 3:18
Martin Sleziak
2,98532128
edited Aug 27 '18 at 3:18
Martin Sleziak
2,98532128
2,98532128
answered Aug 27 '18 at 2:36
Mikko KorhonenMikko Korhonen
1,070914
answered Aug 27 '18 at 2:36
Mikko KorhonenMikko Korhonen
1,070914
answered Aug 27 '18 at 2:36
Mikko KorhonenMikko Korhonen
1,070914
1,070914
$begingroup$
Thanks for your answer! Can we get things beyond order, like generators of G?
$endgroup$
– Sirui Lu
Aug 27 '18 at 2:39
1
$begingroup$
@SiruiLu: Not sure what you mean. If you know $operatornameAut(H)$ and $operatornameHom(H, Z(H))$, you have an explicit description of all the automorphisms of $G$, not just the number of them.
$endgroup$
– Mikko Korhonen
Aug 27 '18 at 2:53
add a comment |
$begingroup$
Thanks for your answer! Can we get things beyond order, like generators of G?
$endgroup$
– Sirui Lu
Aug 27 '18 at 2:39
1
$begingroup$
@SiruiLu: Not sure what you mean. If you know $operatornameAut(H)$ and $operatornameHom(H, Z(H))$, you have an explicit description of all the automorphisms of $G$, not just the number of them.
$endgroup$
– Mikko Korhonen
Aug 27 '18 at 2:53
$begingroup$
Thanks for your answer! Can we get things beyond order, like generators of G?
$endgroup$
– Sirui Lu
Aug 27 '18 at 2:39
$begingroup$
Thanks for your answer! Can we get things beyond order, like generators of G?
$endgroup$
– Sirui Lu
Aug 27 '18 at 2:39
1
1
$begingroup$
@SiruiLu: Not sure what you mean. If you know $operatornameAut(H)$ and $operatornameHom(H, Z(H))$, you have an explicit description of all the automorphisms of $G$, not just the number of them.
$endgroup$
– Mikko Korhonen
Aug 27 '18 at 2:53
$begingroup$
@SiruiLu: Not sure what you mean. If you know $operatornameAut(H)$ and $operatornameHom(H, Z(H))$, you have an explicit description of all the automorphisms of $G$, not just the number of them.
$endgroup$
– Mikko Korhonen
Aug 27 '18 at 2:53
add a comment |
$begingroup$
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) ] ]
answered Sep 15 '18 at 15:08
Stefan KohlStefan Kohl
12.5k956112
$endgroup$
add a comment |
$begingroup$
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) ] ]
answered Sep 15 '18 at 15:08
Stefan KohlStefan Kohl
12.5k956112
$endgroup$
add a comment |
$begingroup$
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) ] ]
answered Sep 15 '18 at 15:08
Stefan KohlStefan Kohl
12.5k956112
$endgroup$
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) ] ]
answered Sep 15 '18 at 15:08
Stefan KohlStefan Kohl
12.5k956112
answered Sep 15 '18 at 15:08
Stefan KohlStefan Kohl
12.5k956112
answered Sep 15 '18 at 15:08
Stefan KohlStefan Kohl
12.5k956112
answered Sep 15 '18 at 15:08
Stefan KohlStefan Kohl
12.5k956112
12.5k956112
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$begingroup$
Is $D_4$ the dihedral group of order $8$?
$endgroup$
– LSpice
Aug 26 '18 at 23:47
$begingroup$
@LSpice Yes it is.
$endgroup$
– Sirui Lu
Aug 26 '18 at 23:47