quotient of normalizer and centralizer is cyclic group
$begingroup$
It's known that if $G/Z(G)$ is a cyclic group, then $G$ is Abelian. Since $G/Z(G)$ is just the special case $H=G$ in the $N/C$ theorem $C_G(H)triangleleft N_G(H)$. I wonder if the below statament is true:
If $H$ is a subgroup of $G$, if $N_G(H)/C_G(H)$ is a cyclic group, then $H$ is Abelian.
I tried using the proof in the original statement, but it didn't work out. If we set $N_G(H)/C_G(H)=langle aC_G(H)rangle$, how can i represent an element in $H$?
abstract-algebra group-theory
edited Jan 7 at 21:14
Bernard
123k741117
asked Jan 7 at 21:04
IdeleIdele
1,020410
$endgroup$
add a comment |
$begingroup$
It's known that if $G/Z(G)$ is a cyclic group, then $G$ is Abelian. Since $G/Z(G)$ is just the special case $H=G$ in the $N/C$ theorem $C_G(H)triangleleft N_G(H)$. I wonder if the below statament is true:
If $H$ is a subgroup of $G$, if $N_G(H)/C_G(H)$ is a cyclic group, then $H$ is Abelian.
I tried using the proof in the original statement, but it didn't work out. If we set $N_G(H)/C_G(H)=langle aC_G(H)rangle$, how can i represent an element in $H$?
abstract-algebra group-theory
edited Jan 7 at 21:14
Bernard
123k741117
asked Jan 7 at 21:04
IdeleIdele
1,020410
$endgroup$
2
$begingroup$
Your hypothesis implies that $H/Z(H)$ is cyclic.
$endgroup$
– Derek Holt
Jan 7 at 21:41
add a comment |
$begingroup$
It's known that if $G/Z(G)$ is a cyclic group, then $G$ is Abelian. Since $G/Z(G)$ is just the special case $H=G$ in the $N/C$ theorem $C_G(H)triangleleft N_G(H)$. I wonder if the below statament is true:
If $H$ is a subgroup of $G$, if $N_G(H)/C_G(H)$ is a cyclic group, then $H$ is Abelian.
I tried using the proof in the original statement, but it didn't work out. If we set $N_G(H)/C_G(H)=langle aC_G(H)rangle$, how can i represent an element in $H$?
abstract-algebra group-theory
edited Jan 7 at 21:14
Bernard
123k741117
asked Jan 7 at 21:04
IdeleIdele
1,020410
$endgroup$
It's known that if $G/Z(G)$ is a cyclic group, then $G$ is Abelian. Since $G/Z(G)$ is just the special case $H=G$ in the $N/C$ theorem $C_G(H)triangleleft N_G(H)$. I wonder if the below statament is true:
If $H$ is a subgroup of $G$, if $N_G(H)/C_G(H)$ is a cyclic group, then $H$ is Abelian.
I tried using the proof in the original statement, but it didn't work out. If we set $N_G(H)/C_G(H)=langle aC_G(H)rangle$, how can i represent an element in $H$?
abstract-algebra group-theory
abstract-algebra group-theory
edited Jan 7 at 21:14
Bernard
123k741117
asked Jan 7 at 21:04
IdeleIdele
1,020410
edited Jan 7 at 21:14
Bernard
123k741117
asked Jan 7 at 21:04
IdeleIdele
1,020410
edited Jan 7 at 21:14
Bernard
123k741117
edited Jan 7 at 21:14
Bernard
123k741117
edited Jan 7 at 21:14
Bernard
123k741117
123k741117
asked Jan 7 at 21:04
IdeleIdele
1,020410
asked Jan 7 at 21:04
IdeleIdele
1,020410
asked Jan 7 at 21:04
IdeleIdele
1,020410
1,020410
2
$begingroup$
Your hypothesis implies that $H/Z(H)$ is cyclic.
$endgroup$
– Derek Holt
Jan 7 at 21:41
add a comment |
2
$begingroup$
Your hypothesis implies that $H/Z(H)$ is cyclic.
$endgroup$
– Derek Holt
Jan 7 at 21:41
2
2
$begingroup$
Your hypothesis implies that $H/Z(H)$ is cyclic.
$endgroup$
– Derek Holt
Jan 7 at 21:41
$begingroup$
Your hypothesis implies that $H/Z(H)$ is cyclic.
$endgroup$
– Derek Holt
Jan 7 at 21:41
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: $H cap C_G(H)=Z(H)$ and $H unlhd N_G(H)$
answered Jan 7 at 22:35
Nicky HeksterNicky Hekster
29.1k63456
$endgroup$
add a comment |
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$begingroup$
Hint: $H cap C_G(H)=Z(H)$ and $H unlhd N_G(H)$
answered Jan 7 at 22:35
Nicky HeksterNicky Hekster
29.1k63456
$endgroup$
add a comment |
$begingroup$
Hint: $H cap C_G(H)=Z(H)$ and $H unlhd N_G(H)$
answered Jan 7 at 22:35
Nicky HeksterNicky Hekster
29.1k63456
$endgroup$
add a comment |
$begingroup$
Hint: $H cap C_G(H)=Z(H)$ and $H unlhd N_G(H)$
answered Jan 7 at 22:35
Nicky HeksterNicky Hekster
29.1k63456
$endgroup$
Hint: $H cap C_G(H)=Z(H)$ and $H unlhd N_G(H)$
answered Jan 7 at 22:35
Nicky HeksterNicky Hekster
29.1k63456
answered Jan 7 at 22:35
Nicky HeksterNicky Hekster
29.1k63456
answered Jan 7 at 22:35
Nicky HeksterNicky Hekster
29.1k63456
answered Jan 7 at 22:35
Nicky HeksterNicky Hekster
29.1k63456
29.1k63456
add a comment |
add a comment |
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$begingroup$
Your hypothesis implies that $H/Z(H)$ is cyclic.
$endgroup$
– Derek Holt
Jan 7 at 21:41