About Sylow subgroup and internal direct product.
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Let $G$ be a finite group and $P$ a Sylow $p$-subgroup of $G$. Let $Q$ be a subgroup of $P$. If $Z(Q)$ is a Sylow $p$-subgroup of $C_G(Q)$, is $C_G(Q)$ a internal direct product of $Z(Q)$ and $O_{p'}(C_G(Q))$?
Notation: $O_{p'}(G)$ is the largest normal subgroup of $G$ of order prime to $p$.
I think the answer is yes, but I can't do it.
It is easy to see that $Z(Q)$ and $O_{p'}(C_G(Q))$ are normal subgroups of $C_G(Q)$, and $Z(Q) cap O_{p'}(C_G(Q)) = 1$.
And I know the following formula :
$$ |AB||A cap B| = |A||B| $$
if $A, B$ are finite subgroups of a group $G$.
How can I prove that $C_G(Q) = Z(Q) O_{p'}(C_G(Q))$?
Thanks!
sylow-theory direct-product
asked Nov 15 at 17:11
Snowbaby
12
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Let $G$ be a finite group and $P$ a Sylow $p$-subgroup of $G$. Let $Q$ be a subgroup of $P$. If $Z(Q)$ is a Sylow $p$-subgroup of $C_G(Q)$, is $C_G(Q)$ a internal direct product of $Z(Q)$ and $O_{p'}(C_G(Q))$?
Notation: $O_{p'}(G)$ is the largest normal subgroup of $G$ of order prime to $p$.
I think the answer is yes, but I can't do it.
It is easy to see that $Z(Q)$ and $O_{p'}(C_G(Q))$ are normal subgroups of $C_G(Q)$, and $Z(Q) cap O_{p'}(C_G(Q)) = 1$.
And I know the following formula :
$$ |AB||A cap B| = |A||B| $$
if $A, B$ are finite subgroups of a group $G$.
How can I prove that $C_G(Q) = Z(Q) O_{p'}(C_G(Q))$?
Thanks!
sylow-theory direct-product
asked Nov 15 at 17:11
Snowbaby
12
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $G$ be a finite group and $P$ a Sylow $p$-subgroup of $G$. Let $Q$ be a subgroup of $P$. If $Z(Q)$ is a Sylow $p$-subgroup of $C_G(Q)$, is $C_G(Q)$ a internal direct product of $Z(Q)$ and $O_{p'}(C_G(Q))$?
Notation: $O_{p'}(G)$ is the largest normal subgroup of $G$ of order prime to $p$.
I think the answer is yes, but I can't do it.
It is easy to see that $Z(Q)$ and $O_{p'}(C_G(Q))$ are normal subgroups of $C_G(Q)$, and $Z(Q) cap O_{p'}(C_G(Q)) = 1$.
And I know the following formula :
$$ |AB||A cap B| = |A||B| $$
if $A, B$ are finite subgroups of a group $G$.
How can I prove that $C_G(Q) = Z(Q) O_{p'}(C_G(Q))$?
Thanks!
sylow-theory direct-product
asked Nov 15 at 17:11
Snowbaby
12
Let $G$ be a finite group and $P$ a Sylow $p$-subgroup of $G$. Let $Q$ be a subgroup of $P$. If $Z(Q)$ is a Sylow $p$-subgroup of $C_G(Q)$, is $C_G(Q)$ a internal direct product of $Z(Q)$ and $O_{p'}(C_G(Q))$?
Notation: $O_{p'}(G)$ is the largest normal subgroup of $G$ of order prime to $p$.
I think the answer is yes, but I can't do it.
It is easy to see that $Z(Q)$ and $O_{p'}(C_G(Q))$ are normal subgroups of $C_G(Q)$, and $Z(Q) cap O_{p'}(C_G(Q)) = 1$.
And I know the following formula :
$$ |AB||A cap B| = |A||B| $$
if $A, B$ are finite subgroups of a group $G$.
How can I prove that $C_G(Q) = Z(Q) O_{p'}(C_G(Q))$?
Thanks!
sylow-theory direct-product
sylow-theory direct-product
asked Nov 15 at 17:11
Snowbaby
12
asked Nov 15 at 17:11
Snowbaby
12
asked Nov 15 at 17:11
Snowbaby
12
asked Nov 15 at 17:11
Snowbaby
12
asked Nov 15 at 17:11
Snowbaby
12
12
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