Let $G$ be an abelian group and a subgroup $Hle G$ . Is $G$ isomorphic to $Htimes (G/H)$.
$begingroup$
Let $G$ be an abelian group and a subgroup $Hle G$ . Is $G$ isomorphic to $Htimes (G/H)$. G can be finite or infinite. Notice this is a bit like Fisrt Homomorphsm Theorem, I tried to prove it like the way done in this theorem , but failed. Is there a counterexample?
abstract-algebra
asked Dec 3 '18 at 6:05
LOISLOIS
3838
$endgroup$
add a comment |
$begingroup$
Let $G$ be an abelian group and a subgroup $Hle G$ . Is $G$ isomorphic to $Htimes (G/H)$. G can be finite or infinite. Notice this is a bit like Fisrt Homomorphsm Theorem, I tried to prove it like the way done in this theorem , but failed. Is there a counterexample?
abstract-algebra
asked Dec 3 '18 at 6:05
LOISLOIS
3838
$endgroup$
add a comment |
$begingroup$
Let $G$ be an abelian group and a subgroup $Hle G$ . Is $G$ isomorphic to $Htimes (G/H)$. G can be finite or infinite. Notice this is a bit like Fisrt Homomorphsm Theorem, I tried to prove it like the way done in this theorem , but failed. Is there a counterexample?
abstract-algebra
asked Dec 3 '18 at 6:05
LOISLOIS
3838
$endgroup$
Let $G$ be an abelian group and a subgroup $Hle G$ . Is $G$ isomorphic to $Htimes (G/H)$. G can be finite or infinite. Notice this is a bit like Fisrt Homomorphsm Theorem, I tried to prove it like the way done in this theorem , but failed. Is there a counterexample?
abstract-algebra
abstract-algebra
asked Dec 3 '18 at 6:05
LOISLOIS
3838
asked Dec 3 '18 at 6:05
LOISLOIS
3838
asked Dec 3 '18 at 6:05
LOISLOIS
3838
asked Dec 3 '18 at 6:05
LOISLOIS
3838
asked Dec 3 '18 at 6:05
LOISLOIS
3838
3838
add a comment |
add a comment |
1 Answer
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$begingroup$
Not in general, no. Let $G=mathbb{Z}_4$ and $H= langle 2 rangle$. Then $H times G/H cong mathbb{Z}_2 times mathbb{Z}_2$ is not isomorphic to $mathbb{Z}_4$.
answered Dec 3 '18 at 6:09
MathematicsStudent1122MathematicsStudent1122
8,60822467
$endgroup$
$begingroup$
what if G is infinite?
$endgroup$
– LOIS
Dec 3 '18 at 8:09
$begingroup$
@LOIS Still no. Just take a similar example (cyclic group).
$endgroup$
– Tobias Kildetoft
Dec 3 '18 at 9:29
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Not in general, no. Let $G=mathbb{Z}_4$ and $H= langle 2 rangle$. Then $H times G/H cong mathbb{Z}_2 times mathbb{Z}_2$ is not isomorphic to $mathbb{Z}_4$.
answered Dec 3 '18 at 6:09
MathematicsStudent1122MathematicsStudent1122
8,60822467
$endgroup$
$begingroup$
what if G is infinite?
$endgroup$
– LOIS
Dec 3 '18 at 8:09
$begingroup$
@LOIS Still no. Just take a similar example (cyclic group).
$endgroup$
– Tobias Kildetoft
Dec 3 '18 at 9:29
add a comment |
$begingroup$
Not in general, no. Let $G=mathbb{Z}_4$ and $H= langle 2 rangle$. Then $H times G/H cong mathbb{Z}_2 times mathbb{Z}_2$ is not isomorphic to $mathbb{Z}_4$.
answered Dec 3 '18 at 6:09
MathematicsStudent1122MathematicsStudent1122
8,60822467
$endgroup$
$begingroup$
what if G is infinite?
$endgroup$
– LOIS
Dec 3 '18 at 8:09
$begingroup$
@LOIS Still no. Just take a similar example (cyclic group).
$endgroup$
– Tobias Kildetoft
Dec 3 '18 at 9:29
add a comment |
$begingroup$
Not in general, no. Let $G=mathbb{Z}_4$ and $H= langle 2 rangle$. Then $H times G/H cong mathbb{Z}_2 times mathbb{Z}_2$ is not isomorphic to $mathbb{Z}_4$.
answered Dec 3 '18 at 6:09
MathematicsStudent1122MathematicsStudent1122
8,60822467
$endgroup$
Not in general, no. Let $G=mathbb{Z}_4$ and $H= langle 2 rangle$. Then $H times G/H cong mathbb{Z}_2 times mathbb{Z}_2$ is not isomorphic to $mathbb{Z}_4$.
answered Dec 3 '18 at 6:09
MathematicsStudent1122MathematicsStudent1122
8,60822467
answered Dec 3 '18 at 6:09
MathematicsStudent1122MathematicsStudent1122
8,60822467
answered Dec 3 '18 at 6:09
MathematicsStudent1122MathematicsStudent1122
8,60822467
answered Dec 3 '18 at 6:09
MathematicsStudent1122MathematicsStudent1122
8,60822467
8,60822467
$begingroup$
what if G is infinite?
$endgroup$
– LOIS
Dec 3 '18 at 8:09
$begingroup$
@LOIS Still no. Just take a similar example (cyclic group).
$endgroup$
– Tobias Kildetoft
Dec 3 '18 at 9:29
add a comment |
$begingroup$
what if G is infinite?
$endgroup$
– LOIS
Dec 3 '18 at 8:09
$begingroup$
@LOIS Still no. Just take a similar example (cyclic group).
$endgroup$
– Tobias Kildetoft
Dec 3 '18 at 9:29
$begingroup$
what if G is infinite?
$endgroup$
– LOIS
Dec 3 '18 at 8:09
$begingroup$
what if G is infinite?
$endgroup$
– LOIS
Dec 3 '18 at 8:09
$begingroup$
@LOIS Still no. Just take a similar example (cyclic group).
$endgroup$
– Tobias Kildetoft
Dec 3 '18 at 9:29
$begingroup$
@LOIS Still no. Just take a similar example (cyclic group).
$endgroup$
– Tobias Kildetoft
Dec 3 '18 at 9:29
add a comment |
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