Cyclic normal subgroups
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I have a question: it is true that if $H_1$ and $H_2$ are cyclic groups that are isomorphic, then $G/H_1$ is isomorphic to $G/H_2$? I know that if I remove the condition "cyclic groups", the given statement is false and there are numerous counterexamples that disproves it, but I don't know if my statement is true and I don't how to create a counterexample or to prove it. If it is true, can you give me a hint about how can this be proven?
For example, I just have shown that $Z_{12}/ langle 2 rangle$ is isomorphic to $Z_{12}/Z_6$ which is isomorphic to $Z_3$ (since both $langle 2 rangle$ and $Z_6$ are isomorphic). How this can be generalized?
group-theory normal-subgroups
edited 2 hours ago
the_fox
2,0711429
asked 4 hours ago
user573497
16419
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up vote
2
down vote
favorite
I have a question: it is true that if $H_1$ and $H_2$ are cyclic groups that are isomorphic, then $G/H_1$ is isomorphic to $G/H_2$? I know that if I remove the condition "cyclic groups", the given statement is false and there are numerous counterexamples that disproves it, but I don't know if my statement is true and I don't how to create a counterexample or to prove it. If it is true, can you give me a hint about how can this be proven?
For example, I just have shown that $Z_{12}/ langle 2 rangle$ is isomorphic to $Z_{12}/Z_6$ which is isomorphic to $Z_3$ (since both $langle 2 rangle$ and $Z_6$ are isomorphic). How this can be generalized?
group-theory normal-subgroups
edited 2 hours ago
the_fox
2,0711429
asked 4 hours ago
user573497
16419
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I have a question: it is true that if $H_1$ and $H_2$ are cyclic groups that are isomorphic, then $G/H_1$ is isomorphic to $G/H_2$? I know that if I remove the condition "cyclic groups", the given statement is false and there are numerous counterexamples that disproves it, but I don't know if my statement is true and I don't how to create a counterexample or to prove it. If it is true, can you give me a hint about how can this be proven?
For example, I just have shown that $Z_{12}/ langle 2 rangle$ is isomorphic to $Z_{12}/Z_6$ which is isomorphic to $Z_3$ (since both $langle 2 rangle$ and $Z_6$ are isomorphic). How this can be generalized?
group-theory normal-subgroups
edited 2 hours ago
the_fox
2,0711429
asked 4 hours ago
user573497
16419
I have a question: it is true that if $H_1$ and $H_2$ are cyclic groups that are isomorphic, then $G/H_1$ is isomorphic to $G/H_2$? I know that if I remove the condition "cyclic groups", the given statement is false and there are numerous counterexamples that disproves it, but I don't know if my statement is true and I don't how to create a counterexample or to prove it. If it is true, can you give me a hint about how can this be proven?
For example, I just have shown that $Z_{12}/ langle 2 rangle$ is isomorphic to $Z_{12}/Z_6$ which is isomorphic to $Z_3$ (since both $langle 2 rangle$ and $Z_6$ are isomorphic). How this can be generalized?
group-theory normal-subgroups
group-theory normal-subgroups
edited 2 hours ago
the_fox
2,0711429
asked 4 hours ago
user573497
16419
edited 2 hours ago
the_fox
2,0711429
asked 4 hours ago
user573497
16419
edited 2 hours ago
the_fox
2,0711429
edited 2 hours ago
the_fox
2,0711429
edited 2 hours ago
the_fox
2,0711429
2,0711429
asked 4 hours ago
user573497
16419
asked 4 hours ago
user573497
16419
asked 4 hours ago
user573497
16419
16419
add a comment |
add a comment |
2 Answers
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4
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No, that's not true. Take $G = C_2 times C_4$ and note that since $G$ is abelian, every subgroup of $G$ is normal in $G$. Let $H = C_2$ be the first direct factor of $G$ and $K$ be the unique subgroup of order $2$ of $C_4$. Obviously, $H cong K$ since both have order $2$, but $G/H cong C_4$ while $G/K cong C_2 times C_2$.
answered 4 hours ago
the_fox
2,0711429
add a comment |
up vote
2
down vote
Did you try cyclic subgroups of the additive group of integers $mathbb{Z}$?
answered 4 hours ago
Bartosz Malman
6881520
But what about integers mod n? I edited my question and I found very interesting that observation about the isomorphism about quotient groups.
– user573497
4 hours ago
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
No, that's not true. Take $G = C_2 times C_4$ and note that since $G$ is abelian, every subgroup of $G$ is normal in $G$. Let $H = C_2$ be the first direct factor of $G$ and $K$ be the unique subgroup of order $2$ of $C_4$. Obviously, $H cong K$ since both have order $2$, but $G/H cong C_4$ while $G/K cong C_2 times C_2$.
answered 4 hours ago
the_fox
2,0711429
add a comment |
up vote
4
down vote
No, that's not true. Take $G = C_2 times C_4$ and note that since $G$ is abelian, every subgroup of $G$ is normal in $G$. Let $H = C_2$ be the first direct factor of $G$ and $K$ be the unique subgroup of order $2$ of $C_4$. Obviously, $H cong K$ since both have order $2$, but $G/H cong C_4$ while $G/K cong C_2 times C_2$.
answered 4 hours ago
the_fox
2,0711429
add a comment |
up vote
4
down vote
up vote
4
down vote
No, that's not true. Take $G = C_2 times C_4$ and note that since $G$ is abelian, every subgroup of $G$ is normal in $G$. Let $H = C_2$ be the first direct factor of $G$ and $K$ be the unique subgroup of order $2$ of $C_4$. Obviously, $H cong K$ since both have order $2$, but $G/H cong C_4$ while $G/K cong C_2 times C_2$.
answered 4 hours ago
the_fox
2,0711429
No, that's not true. Take $G = C_2 times C_4$ and note that since $G$ is abelian, every subgroup of $G$ is normal in $G$. Let $H = C_2$ be the first direct factor of $G$ and $K$ be the unique subgroup of order $2$ of $C_4$. Obviously, $H cong K$ since both have order $2$, but $G/H cong C_4$ while $G/K cong C_2 times C_2$.
answered 4 hours ago
the_fox
2,0711429
answered 4 hours ago
the_fox
2,0711429
answered 4 hours ago
the_fox
2,0711429
answered 4 hours ago
the_fox
2,0711429
2,0711429
add a comment |
add a comment |
up vote
2
down vote
Did you try cyclic subgroups of the additive group of integers $mathbb{Z}$?
answered 4 hours ago
Bartosz Malman
6881520
But what about integers mod n? I edited my question and I found very interesting that observation about the isomorphism about quotient groups.
– user573497
4 hours ago
add a comment |
up vote
2
down vote
Did you try cyclic subgroups of the additive group of integers $mathbb{Z}$?
answered 4 hours ago
Bartosz Malman
6881520
But what about integers mod n? I edited my question and I found very interesting that observation about the isomorphism about quotient groups.
– user573497
4 hours ago
add a comment |
up vote
2
down vote
up vote
2
down vote
Did you try cyclic subgroups of the additive group of integers $mathbb{Z}$?
answered 4 hours ago
Bartosz Malman
6881520
Did you try cyclic subgroups of the additive group of integers $mathbb{Z}$?
answered 4 hours ago
Bartosz Malman
6881520
answered 4 hours ago
Bartosz Malman
6881520
answered 4 hours ago
Bartosz Malman
6881520
answered 4 hours ago
Bartosz Malman
6881520
6881520
But what about integers mod n? I edited my question and I found very interesting that observation about the isomorphism about quotient groups.
– user573497
4 hours ago
add a comment |
But what about integers mod n? I edited my question and I found very interesting that observation about the isomorphism about quotient groups.
– user573497
4 hours ago
But what about integers mod n? I edited my question and I found very interesting that observation about the isomorphism about quotient groups.
– user573497
4 hours ago
But what about integers mod n? I edited my question and I found very interesting that observation about the isomorphism about quotient groups.
– user573497
4 hours ago
add a comment |
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