Question

$\mathbb{Z}_m \times \mathbb Z_n$ is cyclic if and only if $\gcd(m,n)=1$

score 11.4k · Answer 1 · 2013-02-21 16:24:57

11.4k

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$\mathbb{Z}_m \times \mathbb Z_n$ is cyclic if and only if $\gcd(m,n)=1$

Published on 21 Feb 2013 - 16:24

score 4k · Answer 2 · 2013-03-06 16:09:18

4k

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Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$.

Published on 06 Mar 2013 - 16:09

score 57 · Answer 3 · 2020-08-03 15:40:46

57

Views

Direct products inside finite simple groups

Published on 03 Aug 2020 - 15:40

score 56 · Answer 4 · 2020-08-11 04:52:16

Let $A,B,C$, and $D$ be integral domains. If $A\times B$ is isomorphic to $C\times D$, prove that $A$ is isomorphic to $C$ or $D$.

score 2.7k · Answer 5 · 2020-08-12 15:44:34

2.7k

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If $H$ is a subgroup of a finite abelian group $G$, then $G$ has a subgroup that is isomorphic to $G/H$.

Published on 12 Aug 2020 - 15:44

score 19 · Answer 6 · 2020-08-28 23:24:15

19

Views

Prove that $f(g_j) = (e_1, e_2, . . . , g_i, e_(i+1), . . . e_n)$ is a group homomorphism for each i and find the kernel and the image of the function

Published on 28 Aug 2020 - 23:24

score 159 · Answer 7 · 2026-03-26 21:27:14

159

Views

$G/ H$ is the Direct Product of Finite and Infinite Cyclic Groups.

Published on 26 Mar 2026 - 21:27

score 748 · Answer 8 · 2026-03-25 06:00:13

748

Views

Constructing the $Z_2 \times Z_2$ group table

Published on 25 Mar 2026 - 6:00

score 48 · Answer 9 · 2020-09-18 00:39:25

48

Views

Subgroups of product groups

Published on 18 Sep 2020 - 0:39

score 43 · Answer 10 · 2020-09-18 18:05:46

43

Views

For groups $K$ and $H$, $G_1=\{(x,e)|x\in K\} $ and $G_2 =\{(e,y)|y\in H\}$, show that $G_1, G_2\le K\times H$.

Published on 18 Sep 2020 - 18:05

score 54 · Answer 11 · 2020-09-21 09:19:26

54

Views

No direct product is denumerable

Published on 21 Sep 2020 - 9:19

score 183 · Answer 12 · 2020-09-24 04:18:51

183

Views

Proving a criterion for recognizing when a group $G$ is a direct product of 2 groups

Published on 24 Sep 2020 - 4:18

score 51 · Answer 13 · 2020-09-24 17:37:32

51

Views

Proving a criterion for recognizing when a group $G$ is a direct product of 2 groups (continued)

Published on 24 Sep 2020 - 17:37

score 150 · Answer 14 · 2020-09-24 20:12:22

150

Views

The importance of $G \cong K \times H$ is just to create the split short exact sequence.

Published on 24 Sep 2020 - 20:12

score 60 · Answer 15 · 2020-09-29 05:09:39

60

Views

$\mathbb{C}^{*}$ as a direct product of two subgroups. (Hint: polar form)

Published on 29 Sep 2020 - 5:09

List Question

$\mathbb{Z}_m \times \mathbb Z_n$ is cyclic if and only if $\gcd(m,n)=1$

Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$.

Direct products inside finite simple groups

Let $A,B,C$, and $D$ be integral domains. If $A\times B$ is isomorphic to $C\times D$, prove that $A$ is isomorphic to $C$ or $D$.

If $H$ is a subgroup of a finite abelian group $G$, then $G$ has a subgroup that is isomorphic to $G/H$.

Prove that $f(g_j) = (e_1, e_2, . . . , g_i, e_(i+1), . . . e_n)$ is a group homomorphism for each i and find the kernel and the image of the function

$G/ H$ is the Direct Product of Finite and Infinite Cyclic Groups.

Constructing the $Z_2 \times Z_2$ group table

Subgroups of product groups

For groups $K$ and $H$, $G_1=\{(x,e)|x\in K\} $ and $G_2 =\{(e,y)|y\in H\}$, show that $G_1, G_2\le K\times H$.

No direct product is denumerable

Proving a criterion for recognizing when a group $G$ is a direct product of 2 groups

Proving a criterion for recognizing when a group $G$ is a direct product of 2 groups (continued)

The importance of $G \cong K \times H$ is just to create the split short exact sequence.

$\mathbb{C}^{*}$ as a direct product of two subgroups. (Hint: polar form)

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