If $G$ has a nontrivial centre, must every subgroup of index $3$ be normal?
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If a group $G$ has a nontrivial centre, must every subgroup of index $3$ be normal?
$S_3$ yields an example of a group with a non-normal subgroup of index $3$, although it has a trivial centre. Moreover, for finite $G$, it's well-known that if $p$ is the smallest prime dividing $|G|$, then any subgroup of index $p$ is normal. Hence the answer to this question is "yes" if $G$ is a finite group of odd order divisible by $3$.
I'm considering dihedral groups as possible counterexamples, but haven't come up with anything.
group-theory normal-subgroups dihedral-groups
edited Dec 30 '18 at 23:51
Shaun
9,759113684
asked Dec 30 '18 at 23:47
MathematicsStudent1122MathematicsStudent1122
8,98332668
$endgroup$
|
show 2 more comments
$begingroup$
If a group $G$ has a nontrivial centre, must every subgroup of index $3$ be normal?
$S_3$ yields an example of a group with a non-normal subgroup of index $3$, although it has a trivial centre. Moreover, for finite $G$, it's well-known that if $p$ is the smallest prime dividing $|G|$, then any subgroup of index $p$ is normal. Hence the answer to this question is "yes" if $G$ is a finite group of odd order divisible by $3$.
I'm considering dihedral groups as possible counterexamples, but haven't come up with anything.
group-theory normal-subgroups dihedral-groups
edited Dec 30 '18 at 23:51
Shaun
9,759113684
asked Dec 30 '18 at 23:47
MathematicsStudent1122MathematicsStudent1122
8,98332668
$endgroup$
3
$begingroup$
Can’t you fancy up your permutation example into a counterexample by direct products?
$endgroup$
– Randall
Dec 31 '18 at 0:25
$begingroup$
Randall's correct; you can just take $S_3 times C_2$.
$endgroup$
– Qiaochu Yuan
Dec 31 '18 at 0:40
$begingroup$
@Randall Thanks. Feel free to post an answer so I can accept it.
$endgroup$
– MathematicsStudent1122
Dec 31 '18 at 0:45
$begingroup$
@MathematicsStudent1122 nah you go for it. It’s good for you.
$endgroup$
– Randall
Dec 31 '18 at 1:10
$begingroup$
The statement is true however, for every non-trivial 3-group ...
$endgroup$
– Nicky Hekster
Dec 31 '18 at 10:34
|
show 2 more comments
$begingroup$
If a group $G$ has a nontrivial centre, must every subgroup of index $3$ be normal?
$S_3$ yields an example of a group with a non-normal subgroup of index $3$, although it has a trivial centre. Moreover, for finite $G$, it's well-known that if $p$ is the smallest prime dividing $|G|$, then any subgroup of index $p$ is normal. Hence the answer to this question is "yes" if $G$ is a finite group of odd order divisible by $3$.
I'm considering dihedral groups as possible counterexamples, but haven't come up with anything.
group-theory normal-subgroups dihedral-groups
edited Dec 30 '18 at 23:51
Shaun
9,759113684
asked Dec 30 '18 at 23:47
MathematicsStudent1122MathematicsStudent1122
8,98332668
$endgroup$
If a group $G$ has a nontrivial centre, must every subgroup of index $3$ be normal?
$S_3$ yields an example of a group with a non-normal subgroup of index $3$, although it has a trivial centre. Moreover, for finite $G$, it's well-known that if $p$ is the smallest prime dividing $|G|$, then any subgroup of index $p$ is normal. Hence the answer to this question is "yes" if $G$ is a finite group of odd order divisible by $3$.
I'm considering dihedral groups as possible counterexamples, but haven't come up with anything.
group-theory normal-subgroups dihedral-groups
group-theory normal-subgroups dihedral-groups
edited Dec 30 '18 at 23:51
Shaun
9,759113684
asked Dec 30 '18 at 23:47
MathematicsStudent1122MathematicsStudent1122
8,98332668
edited Dec 30 '18 at 23:51
Shaun
9,759113684
asked Dec 30 '18 at 23:47
MathematicsStudent1122MathematicsStudent1122
8,98332668
edited Dec 30 '18 at 23:51
Shaun
9,759113684
edited Dec 30 '18 at 23:51
Shaun
9,759113684
edited Dec 30 '18 at 23:51
Shaun
9,759113684
9,759113684
asked Dec 30 '18 at 23:47
MathematicsStudent1122MathematicsStudent1122
8,98332668
asked Dec 30 '18 at 23:47
MathematicsStudent1122MathematicsStudent1122
8,98332668
asked Dec 30 '18 at 23:47
MathematicsStudent1122MathematicsStudent1122
8,98332668
8,98332668
3
$begingroup$
Can’t you fancy up your permutation example into a counterexample by direct products?
$endgroup$
– Randall
Dec 31 '18 at 0:25
$begingroup$
Randall's correct; you can just take $S_3 times C_2$.
$endgroup$
– Qiaochu Yuan
Dec 31 '18 at 0:40
$begingroup$
@Randall Thanks. Feel free to post an answer so I can accept it.
$endgroup$
– MathematicsStudent1122
Dec 31 '18 at 0:45
$begingroup$
@MathematicsStudent1122 nah you go for it. It’s good for you.
$endgroup$
– Randall
Dec 31 '18 at 1:10
$begingroup$
The statement is true however, for every non-trivial 3-group ...
$endgroup$
– Nicky Hekster
Dec 31 '18 at 10:34
|
show 2 more comments
3
$begingroup$
Can’t you fancy up your permutation example into a counterexample by direct products?
$endgroup$
– Randall
Dec 31 '18 at 0:25
$begingroup$
Randall's correct; you can just take $S_3 times C_2$.
$endgroup$
– Qiaochu Yuan
Dec 31 '18 at 0:40
$begingroup$
@Randall Thanks. Feel free to post an answer so I can accept it.
$endgroup$
– MathematicsStudent1122
Dec 31 '18 at 0:45
$begingroup$
@MathematicsStudent1122 nah you go for it. It’s good for you.
$endgroup$
– Randall
Dec 31 '18 at 1:10
$begingroup$
The statement is true however, for every non-trivial 3-group ...
$endgroup$
– Nicky Hekster
Dec 31 '18 at 10:34
3
3
$begingroup$
Can’t you fancy up your permutation example into a counterexample by direct products?
$endgroup$
– Randall
Dec 31 '18 at 0:25
$begingroup$
Can’t you fancy up your permutation example into a counterexample by direct products?
$endgroup$
– Randall
Dec 31 '18 at 0:25
$begingroup$
Randall's correct; you can just take $S_3 times C_2$.
$endgroup$
– Qiaochu Yuan
Dec 31 '18 at 0:40
$begingroup$
Randall's correct; you can just take $S_3 times C_2$.
$endgroup$
– Qiaochu Yuan
Dec 31 '18 at 0:40
$begingroup$
@Randall Thanks. Feel free to post an answer so I can accept it.
$endgroup$
– MathematicsStudent1122
Dec 31 '18 at 0:45
$begingroup$
@Randall Thanks. Feel free to post an answer so I can accept it.
$endgroup$
– MathematicsStudent1122
Dec 31 '18 at 0:45
$begingroup$
@MathematicsStudent1122 nah you go for it. It’s good for you.
$endgroup$
– Randall
Dec 31 '18 at 1:10
$begingroup$
@MathematicsStudent1122 nah you go for it. It’s good for you.
$endgroup$
– Randall
Dec 31 '18 at 1:10
$begingroup$
The statement is true however, for every non-trivial 3-group ...
$endgroup$
– Nicky Hekster
Dec 31 '18 at 10:34
$begingroup$
The statement is true however, for every non-trivial 3-group ...
$endgroup$
– Nicky Hekster
Dec 31 '18 at 10:34
|
show 2 more comments
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3
$begingroup$
Can’t you fancy up your permutation example into a counterexample by direct products?
$endgroup$
– Randall
Dec 31 '18 at 0:25
$begingroup$
Randall's correct; you can just take $S_3 times C_2$.
$endgroup$
– Qiaochu Yuan
Dec 31 '18 at 0:40
$begingroup$
@Randall Thanks. Feel free to post an answer so I can accept it.
$endgroup$
– MathematicsStudent1122
Dec 31 '18 at 0:45
$begingroup$
@MathematicsStudent1122 nah you go for it. It’s good for you.
$endgroup$
– Randall
Dec 31 '18 at 1:10
$begingroup$
The statement is true however, for every non-trivial 3-group ...
$endgroup$
– Nicky Hekster
Dec 31 '18 at 10:34