Question

Is there a functor $F$ preserving finite direct sums but not split exact sequences, for which $F\mathbb{Z}$ is free and finitely generated?

score 171 · Answer 1 · 2026-03-25 23:58:45

171

Views

Is there a functor $F$ preserving finite direct sums but not split exact sequences, for which $F\mathbb{Z}$ is free and finitely generated?

Published on 25 Mar 2026 - 23:58

score 320 · Answer 2 · 2026-03-26 07:33:54

320

Views

Duality for free abelian groups and for finite abelian groups

Published on 26 Mar 2026 - 7:33

score 65 · Answer 3 · 2026-03-25 23:53:07

65

Views

Does a functor which is additive by arbitrary isomorphism restrict and corestrict to an actually additive functor?

Published on 25 Mar 2026 - 23:53

score 69 · Answer 4 · 2026-03-26 14:34:40

69

Views

Lattice structure of free abelian group on a set $X$

Published on 26 Mar 2026 - 14:34

score 143 · Answer 5 · 2026-04-15 06:59:29

143

Views

Does the Unique Product Property mean that there are only two elements in the set that can create any other in the set, given an operation?

Published on 15 Apr 2026 - 6:59

score 46 · Answer 6 · 2026-03-25 21:22:57

46

Views

Quotient of $\mathbb{Z} \times \mathbb{Z}$ by subgroup generated by $(n,m)$.

Published on 25 Mar 2026 - 21:22

score 156 · Answer 7 · 2026-04-05 18:21:59

156

Views

Free abelian group - exercise

Published on 05 Apr 2026 - 18:21

score 59 · Answer 8 · 2026-03-26 08:23:10

59

Views

Determine what the automorphism group of $\Bbb Z_{7^3}$ looks like.

Published on 26 Mar 2026 - 8:23

score 99 · Answer 9 · 2026-03-26 08:14:53

99

Views

Prove that $A/B$ is cyclic if and only if $\gcd(a,b,c,d) = 1$.

Published on 26 Mar 2026 - 8:14

score 138 · Answer 10 · 2026-04-12 12:12:02

138

Views

The existence of group isomorphism between Euclidean space.

Published on 12 Apr 2026 - 12:12

score 116 · Answer 11 · 2026-04-17 05:08:33

116

Views

A free abelian group and a free group quotient by its commutator group

Published on 17 Apr 2026 - 5:08

score 340 · Answer 12 · 2026-04-17 07:29:16

340

Views

Let $S$ be a generating subset of a free abelian group of rank $k$ s.t. $|S|=k$ , then is the set $S$ linearly independent?

Published on 17 Apr 2026 - 7:29

score 214 · Answer 13 · 2026-04-17 14:50:43

214

Views

$G$ be a free abelian group of rank $k$ , let $S$ be a subset of $G$ that generates $G$ then is it true that $|S| \ge k$?

Published on 17 Apr 2026 - 14:50

score 40 · Answer 14 · 2026-04-15 12:32:12

40

Views

$G$ be a free abelian group of rank $k$ , let $S$ be a subset of $G$ linearly independent over $\mathbb Z$ , then is it true that $|S| \le k$?

Published on 15 Apr 2026 - 12:32

score 406 · Answer 15 · 2026-04-16 23:19:14

406

Views

Prove that any subgroup of a free Abelian group of rank $r$ is free Abelian of rank at most $r$.

Published on 16 Apr 2026 - 23:19

List Question

Is there a functor $F$ preserving finite direct sums but not split exact sequences, for which $F\mathbb{Z}$ is free and finitely generated?

Duality for free abelian groups and for finite abelian groups

Does a functor which is additive by arbitrary isomorphism restrict and corestrict to an actually additive functor?

Lattice structure of free abelian group on a set $X$

Does the Unique Product Property mean that there are only two elements in the set that can create any other in the set, given an operation?

Quotient of $\mathbb{Z} \times \mathbb{Z}$ by subgroup generated by $(n,m)$.

Free abelian group - exercise

Determine what the automorphism group of $\Bbb Z_{7^3}$ looks like.

Prove that $A/B$ is cyclic if and only if $\gcd(a,b,c,d) = 1$.

The existence of group isomorphism between Euclidean space.

A free abelian group and a free group quotient by its commutator group

Let $S$ be a generating subset of a free abelian group of rank $k$ s.t. $|S|=k$ , then is the set $S$ linearly independent?

$G$ be a free abelian group of rank $k$ , let $S$ be a subset of $G$ that generates $G$ then is it true that $|S| \ge k$?

$G$ be a free abelian group of rank $k$ , let $S$ be a subset of $G$ linearly independent over $\mathbb Z$ , then is it true that $|S| \le k$?

Prove that any subgroup of a free Abelian group of rank $r$ is free Abelian of rank at most $r$.

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