For a subgroup $H$ of a finite group $G$ , when does $lvert operatorname{Aut}(H)rvert$ divide $lvert...
$begingroup$
Let $H$ be a subgroup of a finite group $G$. Is it true that $lvert operatorname{Aut}(H)rvert$ divides $lvert operatorname{Aut}(G)rvert$? What if we also assume $G$ is abelian? (I know that $lvert operatorname{Aut}(H)rvert space big| space lvert operatorname{Aut}(G)rvert$ if $G$ is cyclic).
group-theory finite-groups abelian-groups
edited Jan 4 at 3:03
the_fox
2,90231538
$endgroup$
add a comment |
$begingroup$
Let $H$ be a subgroup of a finite group $G$. Is it true that $lvert operatorname{Aut}(H)rvert$ divides $lvert operatorname{Aut}(G)rvert$? What if we also assume $G$ is abelian? (I know that $lvert operatorname{Aut}(H)rvert space big| space lvert operatorname{Aut}(G)rvert$ if $G$ is cyclic).
group-theory finite-groups abelian-groups
edited Jan 4 at 3:03
the_fox
2,90231538
$endgroup$
$begingroup$
This is relevant: mathoverflow.net/questions/9749/…
$endgroup$
– hjhjhj57
Apr 2 '15 at 5:31
add a comment |
$begingroup$
Let $H$ be a subgroup of a finite group $G$. Is it true that $lvert operatorname{Aut}(H)rvert$ divides $lvert operatorname{Aut}(G)rvert$? What if we also assume $G$ is abelian? (I know that $lvert operatorname{Aut}(H)rvert space big| space lvert operatorname{Aut}(G)rvert$ if $G$ is cyclic).
group-theory finite-groups abelian-groups
edited Jan 4 at 3:03
the_fox
2,90231538
$endgroup$
Let $H$ be a subgroup of a finite group $G$. Is it true that $lvert operatorname{Aut}(H)rvert$ divides $lvert operatorname{Aut}(G)rvert$? What if we also assume $G$ is abelian? (I know that $lvert operatorname{Aut}(H)rvert space big| space lvert operatorname{Aut}(G)rvert$ if $G$ is cyclic).
group-theory finite-groups abelian-groups
group-theory finite-groups abelian-groups
edited Jan 4 at 3:03
the_fox
2,90231538
edited Jan 4 at 3:03
the_fox
2,90231538
edited Jan 4 at 3:03
the_fox
2,90231538
edited Jan 4 at 3:03
the_fox
2,90231538
edited Jan 4 at 3:03
the_fox
2,90231538
2,90231538
asked Apr 2 '15 at 4:50
user228168
$begingroup$
This is relevant: mathoverflow.net/questions/9749/…
$endgroup$
– hjhjhj57
Apr 2 '15 at 5:31
add a comment |
$begingroup$
This is relevant: mathoverflow.net/questions/9749/…
$endgroup$
– hjhjhj57
Apr 2 '15 at 5:31
$begingroup$
This is relevant: mathoverflow.net/questions/9749/…
$endgroup$
– hjhjhj57
Apr 2 '15 at 5:31
$begingroup$
This is relevant: mathoverflow.net/questions/9749/…
$endgroup$
– hjhjhj57
Apr 2 '15 at 5:31
add a comment |
1 Answer
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$begingroup$
It's not even true for abelian groups in general. Take $H=C_2times C_2$ as a subgroup of $G=C_4times C_2$. Then $lvert operatorname{Aut}(G)rvert=8$, while $lvert operatorname{Aut}(H)rvert=6$.
edited Jan 4 at 3:06
the_fox
2,90231538
answered Apr 2 '15 at 5:18
verretverret
3,3161923
$endgroup$
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1 Answer
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1 Answer
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active
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votes
$begingroup$
It's not even true for abelian groups in general. Take $H=C_2times C_2$ as a subgroup of $G=C_4times C_2$. Then $lvert operatorname{Aut}(G)rvert=8$, while $lvert operatorname{Aut}(H)rvert=6$.
edited Jan 4 at 3:06
the_fox
2,90231538
answered Apr 2 '15 at 5:18
verretverret
3,3161923
$endgroup$
add a comment |
$begingroup$
It's not even true for abelian groups in general. Take $H=C_2times C_2$ as a subgroup of $G=C_4times C_2$. Then $lvert operatorname{Aut}(G)rvert=8$, while $lvert operatorname{Aut}(H)rvert=6$.
edited Jan 4 at 3:06
the_fox
2,90231538
answered Apr 2 '15 at 5:18
verretverret
3,3161923
$endgroup$
add a comment |
$begingroup$
It's not even true for abelian groups in general. Take $H=C_2times C_2$ as a subgroup of $G=C_4times C_2$. Then $lvert operatorname{Aut}(G)rvert=8$, while $lvert operatorname{Aut}(H)rvert=6$.
edited Jan 4 at 3:06
the_fox
2,90231538
answered Apr 2 '15 at 5:18
verretverret
3,3161923
$endgroup$
It's not even true for abelian groups in general. Take $H=C_2times C_2$ as a subgroup of $G=C_4times C_2$. Then $lvert operatorname{Aut}(G)rvert=8$, while $lvert operatorname{Aut}(H)rvert=6$.
edited Jan 4 at 3:06
the_fox
2,90231538
answered Apr 2 '15 at 5:18
verretverret
3,3161923
edited Jan 4 at 3:06
the_fox
2,90231538
edited Jan 4 at 3:06
the_fox
2,90231538
edited Jan 4 at 3:06
the_fox
2,90231538
2,90231538
answered Apr 2 '15 at 5:18
verretverret
3,3161923
answered Apr 2 '15 at 5:18
verretverret
3,3161923
answered Apr 2 '15 at 5:18
verretverret
3,3161923
3,3161923
add a comment |
add a comment |
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$begingroup$
This is relevant: mathoverflow.net/questions/9749/…
$endgroup$
– hjhjhj57
Apr 2 '15 at 5:31