Counterexamples to $3^{cl(G)}> |G|,$ where $cl(G)$ is the number of conjugacy classes of $G$.
$begingroup$
Let $G$ be a finite group of order $|G|$ and let $cl(G)$ denote the number of conjugacy classes of $G$. Consider the class of groups which satisfy the inequality:
$$3^{cl(G)}> |G|.$$
This class includes, for example, all abelian groups, all symmetric groups, and is closed under taking direct products.
The inequality seems to hold comfortably most of the time but is occasionally tested, e.g., for the Mathieu group $G = M(22)$, which has order $443,520$ but only $12$ conjugacy classes (showing that the $3$
above cannot be replaced by $2$ or $e$).
Question: Are any finite groups known which do not satisfy the inequality?
Thanks
combinatorics inequality finite-groups examples-counterexamples
edited Dec 22 '18 at 4:28
Shaun
9,380113684
asked Sep 9 '15 at 17:29
user2052user2052
1,129613
$endgroup$
add a comment |
$begingroup$
Let $G$ be a finite group of order $|G|$ and let $cl(G)$ denote the number of conjugacy classes of $G$. Consider the class of groups which satisfy the inequality:
$$3^{cl(G)}> |G|.$$
This class includes, for example, all abelian groups, all symmetric groups, and is closed under taking direct products.
The inequality seems to hold comfortably most of the time but is occasionally tested, e.g., for the Mathieu group $G = M(22)$, which has order $443,520$ but only $12$ conjugacy classes (showing that the $3$
above cannot be replaced by $2$ or $e$).
Question: Are any finite groups known which do not satisfy the inequality?
Thanks
combinatorics inequality finite-groups examples-counterexamples
edited Dec 22 '18 at 4:28
Shaun
9,380113684
asked Sep 9 '15 at 17:29
user2052user2052
1,129613
$endgroup$
1
$begingroup$
See also these lower bounds on $cl(G)$, and references here.
$endgroup$
– Dietrich Burde
Sep 9 '15 at 17:48
add a comment |
$begingroup$
Let $G$ be a finite group of order $|G|$ and let $cl(G)$ denote the number of conjugacy classes of $G$. Consider the class of groups which satisfy the inequality:
$$3^{cl(G)}> |G|.$$
This class includes, for example, all abelian groups, all symmetric groups, and is closed under taking direct products.
The inequality seems to hold comfortably most of the time but is occasionally tested, e.g., for the Mathieu group $G = M(22)$, which has order $443,520$ but only $12$ conjugacy classes (showing that the $3$
above cannot be replaced by $2$ or $e$).
Question: Are any finite groups known which do not satisfy the inequality?
Thanks
combinatorics inequality finite-groups examples-counterexamples
edited Dec 22 '18 at 4:28
Shaun
9,380113684
asked Sep 9 '15 at 17:29
user2052user2052
1,129613
$endgroup$
Let $G$ be a finite group of order $|G|$ and let $cl(G)$ denote the number of conjugacy classes of $G$. Consider the class of groups which satisfy the inequality:
$$3^{cl(G)}> |G|.$$
This class includes, for example, all abelian groups, all symmetric groups, and is closed under taking direct products.
The inequality seems to hold comfortably most of the time but is occasionally tested, e.g., for the Mathieu group $G = M(22)$, which has order $443,520$ but only $12$ conjugacy classes (showing that the $3$
above cannot be replaced by $2$ or $e$).
Question: Are any finite groups known which do not satisfy the inequality?
Thanks
combinatorics inequality finite-groups examples-counterexamples
combinatorics inequality finite-groups examples-counterexamples
edited Dec 22 '18 at 4:28
Shaun
9,380113684
asked Sep 9 '15 at 17:29
user2052user2052
1,129613
edited Dec 22 '18 at 4:28
Shaun
9,380113684
asked Sep 9 '15 at 17:29
user2052user2052
1,129613
edited Dec 22 '18 at 4:28
Shaun
9,380113684
edited Dec 22 '18 at 4:28
Shaun
9,380113684
edited Dec 22 '18 at 4:28
Shaun
9,380113684
9,380113684
asked Sep 9 '15 at 17:29
user2052user2052
1,129613
asked Sep 9 '15 at 17:29
user2052user2052
1,129613
asked Sep 9 '15 at 17:29
user2052user2052
1,129613
1,129613
1
$begingroup$
See also these lower bounds on $cl(G)$, and references here.
$endgroup$
– Dietrich Burde
Sep 9 '15 at 17:48
add a comment |
1
$begingroup$
See also these lower bounds on $cl(G)$, and references here.
$endgroup$
– Dietrich Burde
Sep 9 '15 at 17:48
1
1
$begingroup$
See also these lower bounds on $cl(G)$, and references here.
$endgroup$
– Dietrich Burde
Sep 9 '15 at 17:48
$begingroup$
See also these lower bounds on $cl(G)$, and references here.
$endgroup$
– Dietrich Burde
Sep 9 '15 at 17:48
add a comment |
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$begingroup$
See also these lower bounds on $cl(G)$, and references here.
$endgroup$
– Dietrich Burde
Sep 9 '15 at 17:48