Let $G$ be a finite abelian group with identity $e$. Prove that $exists$ an element $x in G$ such that ord y...
$begingroup$
My attempt:
I think somehow we have to show that if $|x|=operatorname{lcm}{|x_i|: x_i in G}$ (as the group is finite abelian) then such an element actually lies in $G-{e}$. But I don't know how to show that such an element indeed lies in $G$.
Thanks for any help!
abstract-algebra group-theory finite-groups
edited Jan 9 at 7:12
CHOUDHARY bhim sen
1589
asked Jan 7 at 4:34
InfinityInfinity
352113
$endgroup$
add a comment |
$begingroup$
My attempt:
I think somehow we have to show that if $|x|=operatorname{lcm}{|x_i|: x_i in G}$ (as the group is finite abelian) then such an element actually lies in $G-{e}$. But I don't know how to show that such an element indeed lies in $G$.
Thanks for any help!
abstract-algebra group-theory finite-groups
edited Jan 9 at 7:12
CHOUDHARY bhim sen
1589
asked Jan 7 at 4:34
InfinityInfinity
352113
$endgroup$
$begingroup$
Do you know the structure theorem for finite Abelian groups?
$endgroup$
– Lord Shark the Unknown
Jan 7 at 4:42
$begingroup$
yeah i do know!
$endgroup$
– Infinity
Jan 7 at 4:43
1
$begingroup$
This is going to be false in the case that $G={e}$, for the trivial fact that any statement of the form "$exists x in varnothing ....$" is always false.
$endgroup$
– MathematicsStudent1122
Jan 7 at 4:50
$begingroup$
Pick an element of maxmial order ...
$endgroup$
– Hagen von Eitzen
Jan 7 at 5:25
add a comment |
$begingroup$
My attempt:
I think somehow we have to show that if $|x|=operatorname{lcm}{|x_i|: x_i in G}$ (as the group is finite abelian) then such an element actually lies in $G-{e}$. But I don't know how to show that such an element indeed lies in $G$.
Thanks for any help!
abstract-algebra group-theory finite-groups
edited Jan 9 at 7:12
CHOUDHARY bhim sen
1589
asked Jan 7 at 4:34
InfinityInfinity
352113
$endgroup$
My attempt:
I think somehow we have to show that if $|x|=operatorname{lcm}{|x_i|: x_i in G}$ (as the group is finite abelian) then such an element actually lies in $G-{e}$. But I don't know how to show that such an element indeed lies in $G$.
Thanks for any help!
abstract-algebra group-theory finite-groups
abstract-algebra group-theory finite-groups
edited Jan 9 at 7:12
CHOUDHARY bhim sen
1589
asked Jan 7 at 4:34
InfinityInfinity
352113
edited Jan 9 at 7:12
CHOUDHARY bhim sen
1589
asked Jan 7 at 4:34
InfinityInfinity
352113
edited Jan 9 at 7:12
CHOUDHARY bhim sen
1589
edited Jan 9 at 7:12
CHOUDHARY bhim sen
1589
edited Jan 9 at 7:12
CHOUDHARY bhim sen
1589
1589
asked Jan 7 at 4:34
InfinityInfinity
352113
asked Jan 7 at 4:34
InfinityInfinity
352113
asked Jan 7 at 4:34
InfinityInfinity
352113
352113
$begingroup$
Do you know the structure theorem for finite Abelian groups?
$endgroup$
– Lord Shark the Unknown
Jan 7 at 4:42
$begingroup$
yeah i do know!
$endgroup$
– Infinity
Jan 7 at 4:43
1
$begingroup$
This is going to be false in the case that $G={e}$, for the trivial fact that any statement of the form "$exists x in varnothing ....$" is always false.
$endgroup$
– MathematicsStudent1122
Jan 7 at 4:50
$begingroup$
Pick an element of maxmial order ...
$endgroup$
– Hagen von Eitzen
Jan 7 at 5:25
add a comment |
$begingroup$
Do you know the structure theorem for finite Abelian groups?
$endgroup$
– Lord Shark the Unknown
Jan 7 at 4:42
$begingroup$
yeah i do know!
$endgroup$
– Infinity
Jan 7 at 4:43
1
$begingroup$
This is going to be false in the case that $G={e}$, for the trivial fact that any statement of the form "$exists x in varnothing ....$" is always false.
$endgroup$
– MathematicsStudent1122
Jan 7 at 4:50
$begingroup$
Pick an element of maxmial order ...
$endgroup$
– Hagen von Eitzen
Jan 7 at 5:25
$begingroup$
Do you know the structure theorem for finite Abelian groups?
$endgroup$
– Lord Shark the Unknown
Jan 7 at 4:42
$begingroup$
Do you know the structure theorem for finite Abelian groups?
$endgroup$
– Lord Shark the Unknown
Jan 7 at 4:42
$begingroup$
yeah i do know!
$endgroup$
– Infinity
Jan 7 at 4:43
$begingroup$
yeah i do know!
$endgroup$
– Infinity
Jan 7 at 4:43
1
1
$begingroup$
This is going to be false in the case that $G={e}$, for the trivial fact that any statement of the form "$exists x in varnothing ....$" is always false.
$endgroup$
– MathematicsStudent1122
Jan 7 at 4:50
$begingroup$
This is going to be false in the case that $G={e}$, for the trivial fact that any statement of the form "$exists x in varnothing ....$" is always false.
$endgroup$
– MathematicsStudent1122
Jan 7 at 4:50
$begingroup$
Pick an element of maxmial order ...
$endgroup$
– Hagen von Eitzen
Jan 7 at 5:25
$begingroup$
Pick an element of maxmial order ...
$endgroup$
– Hagen von Eitzen
Jan 7 at 5:25
add a comment |
1 Answer
1
active
oldest
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$begingroup$
If $G = bigoplus_{1 leq i leq k} mathbb{Z}_{n_i}$ with $n_i | n_{i+1}$, consider the element $(0,0, cdots ,1)$ with order $n_k$. It is clear that $exp G = n_k$.
answered Jan 7 at 4:48
MathematicsStudent1122MathematicsStudent1122
8,98332668
$endgroup$
$begingroup$
Ah okay..Thanks!
$endgroup$
– Infinity
Jan 7 at 4:52
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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active
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votes
$begingroup$
If $G = bigoplus_{1 leq i leq k} mathbb{Z}_{n_i}$ with $n_i | n_{i+1}$, consider the element $(0,0, cdots ,1)$ with order $n_k$. It is clear that $exp G = n_k$.
answered Jan 7 at 4:48
MathematicsStudent1122MathematicsStudent1122
8,98332668
$endgroup$
$begingroup$
Ah okay..Thanks!
$endgroup$
– Infinity
Jan 7 at 4:52
add a comment |
$begingroup$
If $G = bigoplus_{1 leq i leq k} mathbb{Z}_{n_i}$ with $n_i | n_{i+1}$, consider the element $(0,0, cdots ,1)$ with order $n_k$. It is clear that $exp G = n_k$.
answered Jan 7 at 4:48
MathematicsStudent1122MathematicsStudent1122
8,98332668
$endgroup$
$begingroup$
Ah okay..Thanks!
$endgroup$
– Infinity
Jan 7 at 4:52
add a comment |
$begingroup$
If $G = bigoplus_{1 leq i leq k} mathbb{Z}_{n_i}$ with $n_i | n_{i+1}$, consider the element $(0,0, cdots ,1)$ with order $n_k$. It is clear that $exp G = n_k$.
answered Jan 7 at 4:48
MathematicsStudent1122MathematicsStudent1122
8,98332668
$endgroup$
If $G = bigoplus_{1 leq i leq k} mathbb{Z}_{n_i}$ with $n_i | n_{i+1}$, consider the element $(0,0, cdots ,1)$ with order $n_k$. It is clear that $exp G = n_k$.
answered Jan 7 at 4:48
MathematicsStudent1122MathematicsStudent1122
8,98332668
answered Jan 7 at 4:48
MathematicsStudent1122MathematicsStudent1122
8,98332668
answered Jan 7 at 4:48
MathematicsStudent1122MathematicsStudent1122
8,98332668
answered Jan 7 at 4:48
MathematicsStudent1122MathematicsStudent1122
8,98332668
8,98332668
$begingroup$
Ah okay..Thanks!
$endgroup$
– Infinity
Jan 7 at 4:52
add a comment |
$begingroup$
Ah okay..Thanks!
$endgroup$
– Infinity
Jan 7 at 4:52
$begingroup$
Ah okay..Thanks!
$endgroup$
– Infinity
Jan 7 at 4:52
$begingroup$
Ah okay..Thanks!
$endgroup$
– Infinity
Jan 7 at 4:52
add a comment |
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$begingroup$
Do you know the structure theorem for finite Abelian groups?
$endgroup$
– Lord Shark the Unknown
Jan 7 at 4:42
$begingroup$
yeah i do know!
$endgroup$
– Infinity
Jan 7 at 4:43
1
$begingroup$
This is going to be false in the case that $G={e}$, for the trivial fact that any statement of the form "$exists x in varnothing ....$" is always false.
$endgroup$
– MathematicsStudent1122
Jan 7 at 4:50
$begingroup$
Pick an element of maxmial order ...
$endgroup$
– Hagen von Eitzen
Jan 7 at 5:25