Question

What are the conditions for cosets of a certain group to form a group themselves?

score 76 · Answer 1 · 2022-05-21 01:51:12

76

Views

What are the conditions for cosets of a certain group to form a group themselves?

Published on 21 May 2022 - 1:51

score 51 · Answer 2 · 2026-03-29 03:22:26

51

Views

On nonsplit noncentral extension of finite simple groups

Published on 29 Mar 2026 - 3:22

score 362 · Answer 3 · 2022-05-30 23:48:15

362

Views

Quotient groups of the positive real numbers

Published on 30 May 2022 - 23:48

score 115 · Answer 4 · 2022-06-01 02:54:54

115

Views

If $aH$ generates $G/H$ where $G$ is cyclic, then $a$ generates $G$.

Published on 01 Jun 2022 - 2:54

score 164 · Answer 5 · 2022-06-06 01:21:11

164

Views

Prove that $(\mathbb{Q}, +)$ has no subgroups of finite index

Published on 06 Jun 2022 - 1:21

score 45 · Answer 6 · 2022-06-10 19:36:32

45

Views

Question clarification request for: list all the subgroups of $\Bbb Z_{20}/K,$ where $K=\{0,4,8, 12,16\}$

Published on 10 Jun 2022 - 19:36

score 55 · Answer 7 · 2022-06-13 12:17:20

55

Views

The quotient group $(\mathbb{R}\times \mathbb{R},+)/\{(a+b\sqrt{2},a-b\sqrt{2}):a,b\in\mathbb{Z}\}$

Published on 13 Jun 2022 - 12:17

score 95 · Answer 8 · 2022-06-13 17:21:48

95

Views

Let $N\unlhd G$ and let $K$ be any subgroup of $G$ that contains $N$. Then $K\unlhd G$ iff $K/N\unlhd G/N$

Published on 13 Jun 2022 - 17:21

score 50 · Answer 9 · 2022-06-15 20:33:13

50

Views

Prove that multiplication is a well-defined operation on the quotient group $R/\mathcal{J}$ where $\mathcal{J}$ is an ideal and $R$ a ring

Published on 15 Jun 2022 - 20:33

score 83 · Answer 10 · 2026-03-26 11:03:14

83

Views

Let $N \unlhd G$. Then $G/N$ is nilpotent of class $c\in\Bbb{N}$ iff $c$ is the smallest natural number such that $\gamma_c(G) \subset N$

Published on 26 Mar 2026 - 11:03

score 24 · Answer 11 · 2022-06-17 17:17:59

24

Views

Clarification needed for the proof of the theorem: If $T$ is any subgroup of $G/N$, then $T=H/N,$ where $H$ is a subgroup of $G$ that contains $N$

Published on 17 Jun 2022 - 17:17

score 54 · Answer 12 · 2022-06-17 21:29:49

54

Views

Proving representation of the same coset in $\mathbb{Z}[i]/(5)$ iff elements are congruent modulo $5$

Published on 17 Jun 2022 - 21:29

score 97 · Answer 13 · 2022-06-21 22:55:37

97

Views

Given $2$ subgroups $N,K < G$ with $N$ normal in $G$ and $N \cap K = \{e\}$ and $G = NK$. Prove that $G/N$ is isomorphic to $K$

Published on 21 Jun 2022 - 22:55

score 374 · Answer 14 · 2022-06-26 00:28:43

374

Views

Prove $(G_1 \times G_2)/(N_1 \times N_2) \cong G_1/N_1 \times G_2/N_2$.

Published on 26 Jun 2022 - 0:28

score 47 · Answer 15 · 2022-06-26 12:13:20

47

Views

If $H , K \trianglelefteq F_2$ with $F_2/H\cong F_2/K$ then $H=K$

Published on 26 Jun 2022 - 12:13

List Question

What are the conditions for cosets of a certain group to form a group themselves?

On nonsplit noncentral extension of finite simple groups

Quotient groups of the positive real numbers

If $aH$ generates $G/H$ where $G$ is cyclic, then $a$ generates $G$.

Prove that $(\mathbb{Q}, +)$ has no subgroups of finite index

Question clarification request for: list all the subgroups of $\Bbb Z_{20}/K,$ where $K=\{0,4,8, 12,16\}$

The quotient group $(\mathbb{R}\times \mathbb{R},+)/\{(a+b\sqrt{2},a-b\sqrt{2}):a,b\in\mathbb{Z}\}$

Let $N\unlhd G$ and let $K$ be any subgroup of $G$ that contains $N$. Then $K\unlhd G$ iff $K/N\unlhd G/N$

Prove that multiplication is a well-defined operation on the quotient group $R/\mathcal{J}$ where $\mathcal{J}$ is an ideal and $R$ a ring

Let $N \unlhd G$. Then $G/N$ is nilpotent of class $c\in\Bbb{N}$ iff $c$ is the smallest natural number such that $\gamma_c(G) \subset N$

Clarification needed for the proof of the theorem: If $T$ is any subgroup of $G/N$, then $T=H/N,$ where $H$ is a subgroup of $G$ that contains $N$

Proving representation of the same coset in $\mathbb{Z}[i]/(5)$ iff elements are congruent modulo $5$

Given $2$ subgroups $N,K < G$ with $N$ normal in $G$ and $N \cap K = \{e\}$ and $G = NK$. Prove that $G/N$ is isomorphic to $K$

Prove $(G_1 \times G_2)/(N_1 \times N_2) \cong G_1/N_1 \times G_2/N_2$.

If $H , K \trianglelefteq F_2$ with $F_2/H\cong F_2/K$ then $H=K$

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