Show that collection of all prime power elements form a Sylow subgroup
$begingroup$
I was proving the result that every finite abelian group $G$ is isomorphic to direct product of its Sylow subgroups and I have a solution which prove it by just showing that $G$ is internal direct product of $A_{p_1}$, $A_{p_2},ldots,A_{p_k}$ for some primes $p_1$, $p_2,ldots,p_k$,
where
$$A_p={x in Gmid text{order}(x)=p^a,text{ where }a text{ is some positive integer}}$$
I am wondering why $A_p$ is Sylow $p$-subgroup.
group-theory
edited Dec 11 '18 at 20:03
Arturo Magidin
262k34586910
asked Dec 10 '18 at 9:35
IbrahimIbrahim
445
$endgroup$
add a comment |
$begingroup$
I was proving the result that every finite abelian group $G$ is isomorphic to direct product of its Sylow subgroups and I have a solution which prove it by just showing that $G$ is internal direct product of $A_{p_1}$, $A_{p_2},ldots,A_{p_k}$ for some primes $p_1$, $p_2,ldots,p_k$,
where
$$A_p={x in Gmid text{order}(x)=p^a,text{ where }a text{ is some positive integer}}$$
I am wondering why $A_p$ is Sylow $p$-subgroup.
group-theory
edited Dec 11 '18 at 20:03
Arturo Magidin
262k34586910
asked Dec 10 '18 at 9:35
IbrahimIbrahim
445
$endgroup$
$begingroup$
What are your thoughts?
$endgroup$
– jgon
Dec 11 '18 at 18:32
$begingroup$
My question is "my teacher proves the result that every finite abelian group can be written as direct product of its Sylow subgroups and at the end he shows that it is direct product of all Ap's ,now the question arises whether these Ap's are Sylow subgroups?
$endgroup$
– Ibrahim
Dec 12 '18 at 6:00
$begingroup$
Please make one correction 'a' is some non negative integer and not positive as defined in Ap
$endgroup$
– Ibrahim
Dec 12 '18 at 6:01
add a comment |
$begingroup$
I was proving the result that every finite abelian group $G$ is isomorphic to direct product of its Sylow subgroups and I have a solution which prove it by just showing that $G$ is internal direct product of $A_{p_1}$, $A_{p_2},ldots,A_{p_k}$ for some primes $p_1$, $p_2,ldots,p_k$,
where
$$A_p={x in Gmid text{order}(x)=p^a,text{ where }a text{ is some positive integer}}$$
I am wondering why $A_p$ is Sylow $p$-subgroup.
group-theory
edited Dec 11 '18 at 20:03
Arturo Magidin
262k34586910
asked Dec 10 '18 at 9:35
IbrahimIbrahim
445
$endgroup$
I was proving the result that every finite abelian group $G$ is isomorphic to direct product of its Sylow subgroups and I have a solution which prove it by just showing that $G$ is internal direct product of $A_{p_1}$, $A_{p_2},ldots,A_{p_k}$ for some primes $p_1$, $p_2,ldots,p_k$,
where
$$A_p={x in Gmid text{order}(x)=p^a,text{ where }a text{ is some positive integer}}$$
I am wondering why $A_p$ is Sylow $p$-subgroup.
group-theory
group-theory
edited Dec 11 '18 at 20:03
Arturo Magidin
262k34586910
asked Dec 10 '18 at 9:35
IbrahimIbrahim
445
edited Dec 11 '18 at 20:03
Arturo Magidin
262k34586910
asked Dec 10 '18 at 9:35
IbrahimIbrahim
445
edited Dec 11 '18 at 20:03
Arturo Magidin
262k34586910
edited Dec 11 '18 at 20:03
Arturo Magidin
262k34586910
edited Dec 11 '18 at 20:03
Arturo Magidin
262k34586910
262k34586910
asked Dec 10 '18 at 9:35
IbrahimIbrahim
445
asked Dec 10 '18 at 9:35
IbrahimIbrahim
445
asked Dec 10 '18 at 9:35
IbrahimIbrahim
445
445
$begingroup$
What are your thoughts?
$endgroup$
– jgon
Dec 11 '18 at 18:32
$begingroup$
My question is "my teacher proves the result that every finite abelian group can be written as direct product of its Sylow subgroups and at the end he shows that it is direct product of all Ap's ,now the question arises whether these Ap's are Sylow subgroups?
$endgroup$
– Ibrahim
Dec 12 '18 at 6:00
$begingroup$
Please make one correction 'a' is some non negative integer and not positive as defined in Ap
$endgroup$
– Ibrahim
Dec 12 '18 at 6:01
add a comment |
$begingroup$
What are your thoughts?
$endgroup$
– jgon
Dec 11 '18 at 18:32
$begingroup$
My question is "my teacher proves the result that every finite abelian group can be written as direct product of its Sylow subgroups and at the end he shows that it is direct product of all Ap's ,now the question arises whether these Ap's are Sylow subgroups?
$endgroup$
– Ibrahim
Dec 12 '18 at 6:00
$begingroup$
Please make one correction 'a' is some non negative integer and not positive as defined in Ap
$endgroup$
– Ibrahim
Dec 12 '18 at 6:01
$begingroup$
What are your thoughts?
$endgroup$
– jgon
Dec 11 '18 at 18:32
$begingroup$
What are your thoughts?
$endgroup$
– jgon
Dec 11 '18 at 18:32
$begingroup$
My question is "my teacher proves the result that every finite abelian group can be written as direct product of its Sylow subgroups and at the end he shows that it is direct product of all Ap's ,now the question arises whether these Ap's are Sylow subgroups?
$endgroup$
– Ibrahim
Dec 12 '18 at 6:00
$begingroup$
My question is "my teacher proves the result that every finite abelian group can be written as direct product of its Sylow subgroups and at the end he shows that it is direct product of all Ap's ,now the question arises whether these Ap's are Sylow subgroups?
$endgroup$
– Ibrahim
Dec 12 '18 at 6:00
$begingroup$
Please make one correction 'a' is some non negative integer and not positive as defined in Ap
$endgroup$
– Ibrahim
Dec 12 '18 at 6:01
$begingroup$
Please make one correction 'a' is some non negative integer and not positive as defined in Ap
$endgroup$
– Ibrahim
Dec 12 '18 at 6:01
add a comment |
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$begingroup$
What are your thoughts?
$endgroup$
– jgon
Dec 11 '18 at 18:32
$begingroup$
My question is "my teacher proves the result that every finite abelian group can be written as direct product of its Sylow subgroups and at the end he shows that it is direct product of all Ap's ,now the question arises whether these Ap's are Sylow subgroups?
$endgroup$
– Ibrahim
Dec 12 '18 at 6:00
$begingroup$
Please make one correction 'a' is some non negative integer and not positive as defined in Ap
$endgroup$
– Ibrahim
Dec 12 '18 at 6:01