|
11
|
|
edited Nov 19 2010 at 12:20 |
Let $$G=\mathbb{Z}/p_1^{e_1}\times\cdots\times\mathbb{Z}/p_n^{e_n}$$ be any finite abelian group.
What are $G$'s subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If $G=\mathbb{Z}/3\times $G=\mathbb{Z}/3\times \mathbb{Z}/9\times \mathbb{Z}/4\times \mathbb{Z}/8$, mathbb{Z}/8,$$ then I can take $$H=3(\mathbb{Z}/3\times \mathbb{Z}/9)\times 2(\mathbb{Z}/4) \times \mathbb{Z}/8.$$ Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism).
Which are the subgroups $H$ in $G$ for which $G/H$ is primary cyclic?
|
|
|
|
|
10
|
|
edited Nov 19 2010 at 5:06 |
Let $$G=\mathbb{Z}/p_1^{e_1}\times\cdots\times\mathbb{Z}/p_n^{e_n}$$ be any finite abelian group.
What are $G$'s subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If $G=\mathbb{Z}/3\times \mathbb{Z}/9\times \mathbb{Z}/4\times \mathbb{Z}/8$, then I can take $H=3(\mathbb{Z}/3\times $H=3(\mathbb{Z}/3\times \mathbb{Z}/9)\times 2(\mathbb{Z}/4) \times \mathbb{Z}/8$. mathbb{Z}/8.$$ Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism).
Which are the subgroups $H$ in $G$ for which $G/H$ is primary cyclic?
|
|
|
|
|
9
|
|
edited Nov 19 2010 at 3:10 |
Are these all the subgroups Subgroups of a finite abelian group?
Let $G=\mathbb{Z}/p_1^{e_1}\times\cdots\times\mathbb{Z}/p_n^{e_n}$ $G=\mathbb{Z}/p_1^{e_1}\times\cdots\times\mathbb{Z}/p_n^{e_n}$$ be any finite abelian group.
What are $G$'s subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If $G=\mathbb{Z}/3\times \mathbb{Z}/9\times \mathbb{Z}/4\times \mathbb{Z}/8$, then I can take $H=3(\mathbb{Z}/3\times \mathbb{Z}/9)\times 2(\mathbb{Z}/4) \times \mathbb{Z}/8$. Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism).
Which are the subgroups $H$ in $G$ for which $G/H$ is primary cyclic?
|
|
|
|
8
|
|
edited Nov 18 2010 at 17:12 |
Subgroups Are these all the subgroups of an a finite abelian group?
Hello. Let G=Z(p1^e1) x ... x Z(pn^en) $G=\mathbb{Z}/p_1^{e_1}\times\cdots\times\mathbb{Z}/p_n^{e_n}$ be any finite abelian group.
What are G's $G$'s subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If G=Z3 x Z9 x Z4 x Z8, $G=\mathbb{Z}/3\times \mathbb{Z}/9\times \mathbb{Z}/4\times \mathbb{Z}/8$, then I can take H=3(Z3 x Z9$H=3(\mathbb{Z}/3\times \mathbb{Z}/9)\times 2(\mathbb{Z}/4) x (2Z4) x Z8. \times \mathbb{Z}/8$. Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism).
Which are the subgroups H $H$ in G $G$ for which G/H $G/H$ is primary cyclic?
Is there anything else (interesting) to say about the collection of subgroups of an abelian group?
Thank you.
|
|
|
|
|
7
|
|
edited Nov 18 2010 at 14:09 |
Hello.
Let G=Z(p1^e1) x ... x Z(pn^en) be any finite abelian group.
What are G's subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If G=Z3 x Z9 x Z4 x Z8, then I can take H=3(Z3 x Z9) x (2Z4) x Z8. Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism).
Which are the subgroups H in G for which G/H is primary cyclic?
Is there anything else (interesting) to say about the collection of subgroups of an abelian group?
Thank you.
|
|
|
|
|
6
|
|
edited Nov 18 2010 at 14:08 |
Hello.
Let G=Z(p1^e1) x ... x Z(pn^en) be any finite abelian group.
What are G's subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If G=Z3 x Z9 x Z4 x Z8, then I can take H=3(Z3 x Z9) x (2Z4) x Z8. Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism).
Which are the subgroups H in G for which G/H is primary cyclic?
Is there anything else (interesting) to say about the collection of subgroups of an abelian group?
Thank you.
|
|
|
|
|
5
|
|
edited Nov 18 2010 at 13:41 |
Hello. Let G=Z(p1^e1) x ... x Z(pn^en) be any finite abelian group.
What are G's subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If G=Z3 x Z9 x Z4 x Z8, then I can take H=3(Z3 x Z9) x (2Z4) x Z8. Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism).
Which are the subgroups H in G for which G/H is primary cyclic?
Is there anything else (interesting) to say about the collection of subgroups of an abelian group?
Thank you.
|
|
|
|
|
4
|
|
edited Nov 18 2010 at 13:12 |
Hello.
Let G=Z(p1^e1) x ... x Z(pn^en) be any finite abelian group.
What are G's subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If G=Z3 x Z9 x Z4 x Z8, then I can take H=3(Z3 x Z9) x (2Z4) x Z8. Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism).
Which are the subgroups H in G for which G/H is primary cyclic?
Is there anything else (interesting) to say about the collection of subgroups of an abelian group?
Thank you.
|
|
|
|
|
3
|
|
edited Nov 18 2010 at 13:10 |
Hello.
Let G=Z(p1^e1) x ... x Z(pn^en) be any finite abelian group.
What are G's subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If G=Z3 x Z9 x Z4 x Z8, then I can take H=3(Z3 x Z9) x (2Z4) x Z8. Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism).
Which are the subgroups H in G for which G/H is primary cyclic?
Is there anything else (interesting) to say about the collection of subgroups of an abelian group?
Thank you.
|
|
|
|
|
2
|
|
edited Nov 16 2010 at 21:50 |
Hello. Let G=Z(p1^e1) x ... x Z(pn^en) be any finite abelian group.
What are G's subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If G=Z3 x Z9 x Z4 x Z8, then I can take H=3(Z3 x Z9) x (2Z4) x Z8. Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism).
Which are the subgroups H in G for which G/H is primary cyclic?
Is there anything else (interesting) to say about the collection of subgroups of an abelian group?
Thank you.
|
|
|
| |
|
Post Reopened by Mark Sapir, Qiaochu Yuan, HW, Anton Geraschenko♦♦
|
occurred Nov 16 2010 at 21:01
|
|
|
|
|
| |
|
Post Closed as "too localized" by Franz Lemmermeyer, Pete L. Clark, José Figueroa-O'Farrill, HW, Mariano Suárez-Alvarez
|
occurred Nov 15 2010 at 15:06
|
|
|
|
|
|
|
1
|
|
asked Nov 15 2010 at 14:47 uuu
|
Subgroups of an abelian group
Hello.
Let G=Z(p1^e1) x ... x Z(pn^en) be any abelian group.
What are G's subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If G=Z3 x Z9 x Z4 x Z8, then I can take H=3(Z3 x Z9) x (2Z4) x Z8. Do I get all subgroups that way? (I'm interested in all subgroups, not just up-to-isomorphism).
Which are the subgroups H in G for which G/H is primary cyclic?
Is there anything else (interesting) to say about the collection of subgroups of an abelian group?
Thank you.
|
|
|
