Extension of a cyclic group by an abelian group
1
$begingroup$
What is meant by "an extension of a cyclic group of prime order $p$ by an abelian group of odd order relatively prime to $p$"?
Can it be a semidirect product of a cyclic group of order $p^2$ by an abelian group of order $q$, where $p,q$ are distinct primes greater than 2?
finite-groups terminology abelian-groups cyclic-groups group-extensions
edited Dec 22 '18 at 4:47
Shaun
9,060113682
asked Aug 26 '18 at 17:05
Buddhini AngelikaBuddhini Angelika
15910
$endgroup$
add a comment |
1
$begingroup$
What is meant by "an extension of a cyclic group of prime order $p$ by an abelian group of odd order relatively prime to $p$"?
Can it be a semidirect product of a cyclic group of order $p^2$ by an abelian group of order $q$, where $p,q$ are distinct primes greater than 2?
finite-groups terminology abelian-groups cyclic-groups group-extensions
edited Dec 22 '18 at 4:47
Shaun
9,060113682
asked Aug 26 '18 at 17:05
Buddhini AngelikaBuddhini Angelika
15910
$endgroup$
$begingroup$
No it couldn't'; its order must be a multiple of $p$ but not of $p^2$.
$endgroup$
– Lord Shark the Unknown
Aug 26 '18 at 17:07
$begingroup$
Is there a way I can get more information about extensions like that?
$endgroup$
– Buddhini Angelika
Aug 26 '18 at 17:10
1
$begingroup$
The order of the extension is the product of those of the normal subgroup ($p$) and the quotient group (relative to $p$) so it cannot be a multiple of $p^2$.
$endgroup$
– Quang Hoang
Dec 22 '18 at 5:09
add a comment |
1
1
1
2
$begingroup$
What is meant by "an extension of a cyclic group of prime order $p$ by an abelian group of odd order relatively prime to $p$"?
Can it be a semidirect product of a cyclic group of order $p^2$ by an abelian group of order $q$, where $p,q$ are distinct primes greater than 2?
finite-groups terminology abelian-groups cyclic-groups group-extensions
edited Dec 22 '18 at 4:47
Shaun
9,060113682
asked Aug 26 '18 at 17:05
Buddhini AngelikaBuddhini Angelika
15910
$endgroup$
What is meant by "an extension of a cyclic group of prime order $p$ by an abelian group of odd order relatively prime to $p$"?
Can it be a semidirect product of a cyclic group of order $p^2$ by an abelian group of order $q$, where $p,q$ are distinct primes greater than 2?
finite-groups terminology abelian-groups cyclic-groups group-extensions
finite-groups terminology abelian-groups cyclic-groups group-extensions
edited Dec 22 '18 at 4:47
Shaun
9,060113682
asked Aug 26 '18 at 17:05
Buddhini AngelikaBuddhini Angelika
15910
edited Dec 22 '18 at 4:47
Shaun
9,060113682
asked Aug 26 '18 at 17:05
Buddhini AngelikaBuddhini Angelika
15910
edited Dec 22 '18 at 4:47
Shaun
9,060113682
edited Dec 22 '18 at 4:47
Shaun
9,060113682
edited Dec 22 '18 at 4:47
Shaun
9,060113682
9,060113682
asked Aug 26 '18 at 17:05
Buddhini AngelikaBuddhini Angelika
15910
asked Aug 26 '18 at 17:05
Buddhini AngelikaBuddhini Angelika
15910
asked Aug 26 '18 at 17:05
Buddhini AngelikaBuddhini Angelika
15910
15910
$begingroup$
No it couldn't'; its order must be a multiple of $p$ but not of $p^2$.
$endgroup$
– Lord Shark the Unknown
Aug 26 '18 at 17:07
$begingroup$
Is there a way I can get more information about extensions like that?
$endgroup$
– Buddhini Angelika
Aug 26 '18 at 17:10
1
$begingroup$
The order of the extension is the product of those of the normal subgroup ($p$) and the quotient group (relative to $p$) so it cannot be a multiple of $p^2$.
$endgroup$
– Quang Hoang
Dec 22 '18 at 5:09
add a comment |
$begingroup$
No it couldn't'; its order must be a multiple of $p$ but not of $p^2$.
$endgroup$
– Lord Shark the Unknown
Aug 26 '18 at 17:07
$begingroup$
Is there a way I can get more information about extensions like that?
$endgroup$
– Buddhini Angelika
Aug 26 '18 at 17:10
1
$begingroup$
The order of the extension is the product of those of the normal subgroup ($p$) and the quotient group (relative to $p$) so it cannot be a multiple of $p^2$.
$endgroup$
– Quang Hoang
Dec 22 '18 at 5:09
$begingroup$
No it couldn't'; its order must be a multiple of $p$ but not of $p^2$.
$endgroup$
– Lord Shark the Unknown
Aug 26 '18 at 17:07
$begingroup$
No it couldn't'; its order must be a multiple of $p$ but not of $p^2$.
$endgroup$
– Lord Shark the Unknown
Aug 26 '18 at 17:07
$begingroup$
Is there a way I can get more information about extensions like that?
$endgroup$
– Buddhini Angelika
Aug 26 '18 at 17:10
$begingroup$
Is there a way I can get more information about extensions like that?
$endgroup$
– Buddhini Angelika
Aug 26 '18 at 17:10
1
1
$begingroup$
The order of the extension is the product of those of the normal subgroup ($p$) and the quotient group (relative to $p$) so it cannot be a multiple of $p^2$.
$endgroup$
– Quang Hoang
Dec 22 '18 at 5:09
$begingroup$
The order of the extension is the product of those of the normal subgroup ($p$) and the quotient group (relative to $p$) so it cannot be a multiple of $p^2$.
$endgroup$
– Quang Hoang
Dec 22 '18 at 5:09
add a comment |
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$begingroup$
No it couldn't'; its order must be a multiple of $p$ but not of $p^2$.
$endgroup$
– Lord Shark the Unknown
Aug 26 '18 at 17:07
$begingroup$
Is there a way I can get more information about extensions like that?
$endgroup$
– Buddhini Angelika
Aug 26 '18 at 17:10
1
$begingroup$
The order of the extension is the product of those of the normal subgroup ($p$) and the quotient group (relative to $p$) so it cannot be a multiple of $p^2$.
$endgroup$
– Quang Hoang
Dec 22 '18 at 5:09