Mixing
Prove that an Abel group whose order doesn’t include square factor is a cyclic group. [closed]
“Prove that an Abel group whose order doesn’t include square factor is a cyclic group.”
How to prove this?
If $G$ is a group which fulfills this condition, can we say
$|G|=p_1p_2...p_n$ (each $p_i$ is different prime number) ?
group-theory
edited Nov 20 '18 at 13:26
amWhy
192k28224439
asked Nov 20 '18 at 13:15
saki
296
closed as off-topic by Shaun, Christopher, Adrian Keister, Dietrich Burde, Derek Holt Nov 20 '18 at 14:14
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, Christopher, Adrian Keister, Dietrich Burde, Derek Holt
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
“Prove that an Abel group whose order doesn’t include square factor is a cyclic group.”
How to prove this?
If $G$ is a group which fulfills this condition, can we say
$|G|=p_1p_2...p_n$ (each $p_i$ is different prime number) ?
group-theory
edited Nov 20 '18 at 13:26
amWhy
192k28224439
asked Nov 20 '18 at 13:15
saki
296
closed as off-topic by Shaun, Christopher, Adrian Keister, Dietrich Burde, Derek Holt Nov 20 '18 at 14:14
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, Christopher, Adrian Keister, Dietrich Burde, Derek Holt
If this question can be reworded to fit the rules in the help center, please edit the question.
1
What do you know about finite abelian groups?
– Arthur
Nov 20 '18 at 13:21
Welcome to Maths SX! Do you know the Chinese remainder theorem?
– Bernard
Nov 20 '18 at 13:21
add a comment |
“Prove that an Abel group whose order doesn’t include square factor is a cyclic group.”
How to prove this?
If $G$ is a group which fulfills this condition, can we say
$|G|=p_1p_2...p_n$ (each $p_i$ is different prime number) ?
group-theory
edited Nov 20 '18 at 13:26
amWhy
192k28224439
asked Nov 20 '18 at 13:15
saki
296
“Prove that an Abel group whose order doesn’t include square factor is a cyclic group.”
How to prove this?
If $G$ is a group which fulfills this condition, can we say
$|G|=p_1p_2...p_n$ (each $p_i$ is different prime number) ?
group-theory
group-theory
edited Nov 20 '18 at 13:26
amWhy
192k28224439
asked Nov 20 '18 at 13:15
saki
296
edited Nov 20 '18 at 13:26
amWhy
192k28224439
asked Nov 20 '18 at 13:15
saki
296
edited Nov 20 '18 at 13:26
amWhy
192k28224439
edited Nov 20 '18 at 13:26
amWhy
192k28224439
edited Nov 20 '18 at 13:26
amWhy
192k28224439
192k28224439
asked Nov 20 '18 at 13:15
saki
296
asked Nov 20 '18 at 13:15
saki
296
asked Nov 20 '18 at 13:15
saki
296
296
closed as off-topic by Shaun, Christopher, Adrian Keister, Dietrich Burde, Derek Holt Nov 20 '18 at 14:14
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, Christopher, Adrian Keister, Dietrich Burde, Derek Holt
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Shaun, Christopher, Adrian Keister, Dietrich Burde, Derek Holt Nov 20 '18 at 14:14
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, Christopher, Adrian Keister, Dietrich Burde, Derek Holt
If this question can be reworded to fit the rules in the help center, please edit the question.
1
What do you know about finite abelian groups?
– Arthur
Nov 20 '18 at 13:21
Welcome to Maths SX! Do you know the Chinese remainder theorem?
– Bernard
Nov 20 '18 at 13:21
add a comment |
1
What do you know about finite abelian groups?
– Arthur
Nov 20 '18 at 13:21
Welcome to Maths SX! Do you know the Chinese remainder theorem?
– Bernard
Nov 20 '18 at 13:21
1
1
What do you know about finite abelian groups?
– Arthur
Nov 20 '18 at 13:21
What do you know about finite abelian groups?
– Arthur
Nov 20 '18 at 13:21
Welcome to Maths SX! Do you know the Chinese remainder theorem?
– Bernard
Nov 20 '18 at 13:21
Welcome to Maths SX! Do you know the Chinese remainder theorem?
– Bernard
Nov 20 '18 at 13:21
add a comment |
1 Answer
1
active
oldest
votes
Hint: By Cauchy's theorem, there is an element $g_p in G$ of order $p$ for each prime $p$ dividing the order of $G$. What is the order of the product of the $g_p$?
answered Nov 20 '18 at 13:23
lhf
163k10167386
The order of the product of the gp is p? So we can say every subgroup of G is circlic group and therefore G is circlic group?
– saki
Nov 20 '18 at 20:58
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Hint: By Cauchy's theorem, there is an element $g_p in G$ of order $p$ for each prime $p$ dividing the order of $G$. What is the order of the product of the $g_p$?
answered Nov 20 '18 at 13:23
lhf
163k10167386
The order of the product of the gp is p? So we can say every subgroup of G is circlic group and therefore G is circlic group?
– saki
Nov 20 '18 at 20:58
add a comment |
Hint: By Cauchy's theorem, there is an element $g_p in G$ of order $p$ for each prime $p$ dividing the order of $G$. What is the order of the product of the $g_p$?
answered Nov 20 '18 at 13:23
lhf
163k10167386
The order of the product of the gp is p? So we can say every subgroup of G is circlic group and therefore G is circlic group?
– saki
Nov 20 '18 at 20:58
add a comment |
Hint: By Cauchy's theorem, there is an element $g_p in G$ of order $p$ for each prime $p$ dividing the order of $G$. What is the order of the product of the $g_p$?
answered Nov 20 '18 at 13:23
lhf
163k10167386
Hint: By Cauchy's theorem, there is an element $g_p in G$ of order $p$ for each prime $p$ dividing the order of $G$. What is the order of the product of the $g_p$?
answered Nov 20 '18 at 13:23
lhf
163k10167386
edited Nov 21 '18 at 0:20
answered Nov 20 '18 at 13:23
lhf
163k10167386
answered Nov 20 '18 at 13:23
lhf
163k10167386
answered Nov 20 '18 at 13:23
lhf
163k10167386
163k10167386
The order of the product of the gp is p? So we can say every subgroup of G is circlic group and therefore G is circlic group?
– saki
Nov 20 '18 at 20:58
add a comment |
The order of the product of the gp is p? So we can say every subgroup of G is circlic group and therefore G is circlic group?
– saki
Nov 20 '18 at 20:58
The order of the product of the gp is p? So we can say every subgroup of G is circlic group and therefore G is circlic group?
– saki
Nov 20 '18 at 20:58
The order of the product of the gp is p? So we can say every subgroup of G is circlic group and therefore G is circlic group?
– saki
Nov 20 '18 at 20:58
add a comment |
1
What do you know about finite abelian groups?
– Arthur
Nov 20 '18 at 13:21
Welcome to Maths SX! Do you know the Chinese remainder theorem?
– Bernard
Nov 20 '18 at 13:21