In a group of order $m p^n$ for $p$ prime, if $k<n$, is there an element of order $p^k$? [duplicate]
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This question already has an answer here:
Existence of elements of order $p^k$ in a finite group
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Let $G$ a group of order $mp^n$ where $p$ is prime. Let $kleq n$. Is there an element of order $p^k$ ?
Since $p$ divide $|G|$, by Cauchy theorem, there is $gin G$ s.t. $g$ has order $p$. I can't do better. I tried to use the fact that there is a $p-$Sylow group, but it just confirm the fact that there is an element of order $p$, not of order $p^k$.
abstract-algebra group-theory finite-groups sylow-theory
edited Dec 13 '18 at 19:55
Shaun
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asked Dec 13 '18 at 19:25
user623855user623855
1457
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marked as duplicate by Dietrich Burde abstract-algebra
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Dec 13 '18 at 19:43
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$begingroup$
This question already has an answer here:
Existence of elements of order $p^k$ in a finite group
3 answers
Let $G$ a group of order $mp^n$ where $p$ is prime. Let $kleq n$. Is there an element of order $p^k$ ?
Since $p$ divide $|G|$, by Cauchy theorem, there is $gin G$ s.t. $g$ has order $p$. I can't do better. I tried to use the fact that there is a $p-$Sylow group, but it just confirm the fact that there is an element of order $p$, not of order $p^k$.
abstract-algebra group-theory finite-groups sylow-theory
edited Dec 13 '18 at 19:55
Shaun
9,310113684
asked Dec 13 '18 at 19:25
user623855user623855
1457
$endgroup$
marked as duplicate by Dietrich Burde abstract-algebra
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Dec 13 '18 at 19:43
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$begingroup$
This question already has an answer here:
Existence of elements of order $p^k$ in a finite group
3 answers
Let $G$ a group of order $mp^n$ where $p$ is prime. Let $kleq n$. Is there an element of order $p^k$ ?
Since $p$ divide $|G|$, by Cauchy theorem, there is $gin G$ s.t. $g$ has order $p$. I can't do better. I tried to use the fact that there is a $p-$Sylow group, but it just confirm the fact that there is an element of order $p$, not of order $p^k$.
abstract-algebra group-theory finite-groups sylow-theory
edited Dec 13 '18 at 19:55
Shaun
9,310113684
asked Dec 13 '18 at 19:25
user623855user623855
1457
$endgroup$
This question already has an answer here:
Existence of elements of order $p^k$ in a finite group
3 answers
Let $G$ a group of order $mp^n$ where $p$ is prime. Let $kleq n$. Is there an element of order $p^k$ ?
Since $p$ divide $|G|$, by Cauchy theorem, there is $gin G$ s.t. $g$ has order $p$. I can't do better. I tried to use the fact that there is a $p-$Sylow group, but it just confirm the fact that there is an element of order $p$, not of order $p^k$.
This question already has an answer here:
Existence of elements of order $p^k$ in a finite group
3 answers
abstract-algebra group-theory finite-groups sylow-theory
abstract-algebra group-theory finite-groups sylow-theory
edited Dec 13 '18 at 19:55
Shaun
9,310113684
asked Dec 13 '18 at 19:25
user623855user623855
1457
edited Dec 13 '18 at 19:55
Shaun
9,310113684
asked Dec 13 '18 at 19:25
user623855user623855
1457
edited Dec 13 '18 at 19:55
Shaun
9,310113684
edited Dec 13 '18 at 19:55
Shaun
9,310113684
edited Dec 13 '18 at 19:55
Shaun
9,310113684
9,310113684
asked Dec 13 '18 at 19:25
user623855user623855
1457
asked Dec 13 '18 at 19:25
user623855user623855
1457
asked Dec 13 '18 at 19:25
user623855user623855
1457
1457
marked as duplicate by Dietrich Burde abstract-algebra
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Dec 13 '18 at 19:43
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1 Answer
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No, for instance when $G=mathbb{F}_p^3$, every element has order $p$.
answered Dec 13 '18 at 19:28
MindlackMindlack
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You gave the "same" answer 9 hours ago here.
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– Dietrich Burde
Dec 13 '18 at 19:44
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1 Answer
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1 Answer
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$begingroup$
No, for instance when $G=mathbb{F}_p^3$, every element has order $p$.
answered Dec 13 '18 at 19:28
MindlackMindlack
4,780210
$endgroup$
1
$begingroup$
You gave the "same" answer 9 hours ago here.
$endgroup$
– Dietrich Burde
Dec 13 '18 at 19:44
add a comment |
$begingroup$
No, for instance when $G=mathbb{F}_p^3$, every element has order $p$.
answered Dec 13 '18 at 19:28
MindlackMindlack
4,780210
$endgroup$
1
$begingroup$
You gave the "same" answer 9 hours ago here.
$endgroup$
– Dietrich Burde
Dec 13 '18 at 19:44
add a comment |
$begingroup$
No, for instance when $G=mathbb{F}_p^3$, every element has order $p$.
answered Dec 13 '18 at 19:28
MindlackMindlack
4,780210
$endgroup$
No, for instance when $G=mathbb{F}_p^3$, every element has order $p$.
answered Dec 13 '18 at 19:28
MindlackMindlack
4,780210
answered Dec 13 '18 at 19:28
MindlackMindlack
4,780210
answered Dec 13 '18 at 19:28
MindlackMindlack
4,780210
answered Dec 13 '18 at 19:28
MindlackMindlack
4,780210
4,780210
1
$begingroup$
You gave the "same" answer 9 hours ago here.
$endgroup$
– Dietrich Burde
Dec 13 '18 at 19:44
add a comment |
1
$begingroup$
You gave the "same" answer 9 hours ago here.
$endgroup$
– Dietrich Burde
Dec 13 '18 at 19:44
1
1
$begingroup$
You gave the "same" answer 9 hours ago here.
$endgroup$
– Dietrich Burde
Dec 13 '18 at 19:44
$begingroup$
You gave the "same" answer 9 hours ago here.
$endgroup$
– Dietrich Burde
Dec 13 '18 at 19:44
add a comment |
