Question

If $N_i \cap N_1N_2\ldots N_{i-1}N_{i-2}\ldots N_k=\left<e\right>$, with each $N_j \unlhd G$, $N_i\cap (N_1N_2\ldots N_{i-1})=\left<e\right>.$

score 54 · Answer 1 · 2022-03-16 19:13:44

54

Views

If $N_i \cap N_1N_2\ldots N_{i-1}N_{i-2}\ldots N_k=\left<e\right>$, with each $N_j \unlhd G$, $N_i\cap (N_1N_2\ldots N_{i-1})=\left<e\right>.$

Published on 16 Mar 2022 - 19:13

score 166 · Answer 2 · 2022-03-18 13:47:22

166

Views

Why is the direct product $\mathbb{R}\times\mathbb{R}$ undefined when $\mathbb{R}$ is viewed as a field?

Published on 18 Mar 2022 - 13:47

score 108 · Answer 3 · 2022-03-21 01:30:04

108

Views

In the category of fields, is $k\times k=k$ if the direct product is well-defined (in the category of fields)?

Published on 21 Mar 2022 - 1:30

score 47 · Answer 4 · 2022-03-23 23:10:09

47

Views

Factors of Direct Composite of Subsemigroups are in fact Normal Subgroups

Published on 23 Mar 2022 - 23:10

score 122 · Answer 5 · 2022-03-24 10:16:06

122

Views

If $S$ is a simple subnormal subgroup of $G$. Prove that if $S$ is nonabelian then $S^G$ is a direct product of simple groups isomorphic to $S$.

Published on 24 Mar 2022 - 10:16

score 411 · Answer 6 · 2022-03-24 20:29:46

411

Views

If $G,H$ are finite groups, then $G\times G\cong H \times H$ implies $G \cong H$

Published on 24 Mar 2022 - 20:29

score 52 · Answer 7 · 2022-03-26 09:16:35

52

Views

Definition of Internal Group Direct Product - how does it follow that $(a,b) \circ (c,d) = (a \circ c, b \circ d)$?

Published on 26 Mar 2022 - 9:16

score 40 · Answer 8 · 2026-03-28 13:21:35

40

Views

Extending a Property of a p-group by Direct Product

Published on 28 Mar 2026 - 13:21

score 60 · Answer 9 · 2022-04-01 02:19:16

60

Views

Normal Subgroup of Direct Product Implies Each Component Normal?

Published on 01 Apr 2022 - 2:19

score 185 · Answer 10 · 2022-04-01 11:55:14

185

Views

Minimal normal subgroups of Product of simple groups

Published on 01 Apr 2022 - 11:55

score 157 · Answer 11 · 2022-04-02 05:07:24

157

Views

Let $G,G'$ be finite groups of orders $m,n$ respectively. What is the order of $G×G'$?

Published on 02 Apr 2022 - 5:07

score 36 · Answer 12 · 2022-04-18 21:49:20

36

Views

Given groups $G_1, G_2$, if $K$ is a subgroup of $G_1 \times G_2$, are there $H_1\leq G_1$ and $H_2\leq G_2$ such that $K=H_1\times H_2$?

Published on 18 Apr 2022 - 21:49

score 47 · Answer 13 · 2022-04-18 22:44:24

47

Views

Let $K=\{(g,g):g \in G\}$ where $G$ is a group. There are no subgroups $H_1,H_2$ of $G$ such that $K=H_1\times H_2$

Published on 18 Apr 2022 - 22:44

score 179 · Answer 14 · 2026-03-25 17:21:42

179

Views

Let $G$ be a finite group and $M$ be a maximal subgroup of $G$. If $G = Z(G)M$, then $M$ is normal in $G$

Published on 25 Mar 2026 - 17:21

score 409 · Answer 15 · 2022-05-06 12:21:15

409

Views

What is the difference between the cartesian product and direct sum of vectors?

Published on 06 May 2022 - 12:21

List Question

If $N_i \cap N_1N_2\ldots N_{i-1}N_{i-2}\ldots N_k=\left<e\right>$, with each $N_j \unlhd G$, $N_i\cap (N_1N_2\ldots N_{i-1})=\left<e\right>.$

Why is the direct product $\mathbb{R}\times\mathbb{R}$ undefined when $\mathbb{R}$ is viewed as a field?

In the category of fields, is $k\times k=k$ if the direct product is well-defined (in the category of fields)?

Factors of Direct Composite of Subsemigroups are in fact Normal Subgroups

If $S$ is a simple subnormal subgroup of $G$. Prove that if $S$ is nonabelian then $S^G$ is a direct product of simple groups isomorphic to $S$.

If $G,H$ are finite groups, then $G\times G\cong H \times H$ implies $G \cong H$

Definition of Internal Group Direct Product - how does it follow that $(a,b) \circ (c,d) = (a \circ c, b \circ d)$?

Extending a Property of a p-group by Direct Product

Normal Subgroup of Direct Product Implies Each Component Normal?

Minimal normal subgroups of Product of simple groups

Let $G,G'$ be finite groups of orders $m,n$ respectively. What is the order of $G×G'$?

Given groups $G_1, G_2$, if $K$ is a subgroup of $G_1 \times G_2$, are there $H_1\leq G_1$ and $H_2\leq G_2$ such that $K=H_1\times H_2$?

Let $K=\{(g,g):g \in G\}$ where $G$ is a group. There are no subgroups $H_1,H_2$ of $G$ such that $K=H_1\times H_2$

Let $G$ be a finite group and $M$ be a maximal subgroup of $G$. If $G = Z(G)M$, then $M$ is normal in $G$

What is the difference between the cartesian product and direct sum of vectors?

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