Question

Is it true that $G\cong\bigoplus_{\alpha\in A}\langle g_\alpha\rangle$ implies $G=\langle g_{\alpha}\mid \alpha\in A\rangle$?

score 42 · Answer 1 · 2026-04-16 02:47:15

42

Views

Is it true that $G\cong\bigoplus_{\alpha\in A}\langle g_\alpha\rangle$ implies $G=\langle g_{\alpha}\mid \alpha\in A\rangle$?

Published on 16 Apr 2026 - 2:47

score 340 · Answer 2 · 2026-04-14 06:49:18

340

Views

Free abelian groups and free groups

Published on 14 Apr 2026 - 6:49

score 183 · Answer 3 · 2026-04-15 11:04:11

183

Views

Primary Decomposition of group $G=\mathbb{Z}^{3}/N$ where $N\leq G$

Published on 15 Apr 2026 - 11:04

score 344 · Answer 4 · 2026-04-16 03:35:18

344

Views

Basis for free Abelian group $A_{n}$ of rank $n$

Published on 16 Apr 2026 - 3:35

score 196 · Answer 5 · 2026-04-14 20:43:27

196

Views

Matrix obtained by sequence of elementary transformations is equal to the original coordinate matrix multiplied by some matrices in $GL(n,\mathbb{Z})$

Published on 14 Apr 2026 - 20:43

score 510 · Answer 6 · 2026-04-17 01:00:43

510

Views

$X$ is a basis for free abelian group $A_{n}$ if and only if $\det (M) = \pm 1$

Published on 17 Apr 2026 - 1:00

score 687 · Answer 7 · 2026-04-16 14:41:34

687

Views

Want to prove $Aut(A_{n})\simeq GL(n, \mathbb{Z})$ where $A_{n}$ is a free abelian group of rank $n$

Published on 16 Apr 2026 - 14:41

score 756 · Answer 8 · 2026-04-15 11:36:24

756

Views

Prove that a free abelian group $A_{n}$ is finitely generated if and only if its rank $n$ is finite

Published on 15 Apr 2026 - 11:36

score 331 · Answer 9 · 2026-04-11 09:46:47

331

Views

Grothendieck Group - why not the free abelian group?

Published on 11 Apr 2026 - 9:46

score 170 · Answer 10 · 2026-04-16 03:53:00

170

Views

Lang's Theorem 7.3 wording, what does he mean?

Published on 16 Apr 2026 - 3:53

score 92 · Answer 11 · 2026-04-12 06:04:01

92

Views

Obtaining an isomorphism from a surjective homomorphism between abelian groups

Published on 12 Apr 2026 - 6:04

score 32 · Answer 12 · 2026-04-12 00:38:36

32

Views

Dual group of the torus

Published on 12 Apr 2026 - 0:38

score 51 · Answer 13 · 2026-04-15 05:26:44

51

Views

Construction of Free abelian groups on Massey Book

Published on 15 Apr 2026 - 5:26

score 121 · Answer 14 · 2026-03-26 11:18:02

121

Views

Proof verification of decomposition of abelian group into torsion and free groups

Published on 26 Mar 2026 - 11:18

score 53 · Answer 15 · 2026-04-13 21:48:37

53

Views

Given a non-singular rational matrix $Q \in \mathbb Q^{n\times n}$, what is $Q(\mathbb Z^{n}) \cap \mathbb Z^{n}$?

Published on 13 Apr 2026 - 21:48

List Question

Is it true that $G\cong\bigoplus_{\alpha\in A}\langle g_\alpha\rangle$ implies $G=\langle g_{\alpha}\mid \alpha\in A\rangle$?

Free abelian groups and free groups

Primary Decomposition of group $G=\mathbb{Z}^{3}/N$ where $N\leq G$

Basis for free Abelian group $A_{n}$ of rank $n$

Matrix obtained by sequence of elementary transformations is equal to the original coordinate matrix multiplied by some matrices in $GL(n,\mathbb{Z})$

$X$ is a basis for free abelian group $A_{n}$ if and only if $\det (M) = \pm 1$

Want to prove $Aut(A_{n})\simeq GL(n, \mathbb{Z})$ where $A_{n}$ is a free abelian group of rank $n$

Prove that a free abelian group $A_{n}$ is finitely generated if and only if its rank $n$ is finite

Grothendieck Group - why not the free abelian group?

Lang's Theorem 7.3 wording, what does he mean?

Obtaining an isomorphism from a surjective homomorphism between abelian groups

Dual group of the torus

Construction of Free abelian groups on Massey Book

Proof verification of decomposition of abelian group into torsion and free groups

Given a non-singular rational matrix $Q \in \mathbb Q^{n\times n}$, what is $Q(\mathbb Z^{n}) \cap \mathbb Z^{n}$?

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