Question

Proving that quotient subgroups of congruent subgroups of modular group are finite.

score 75 · Answer 1 · 2019-12-21 11:26:18

75

Views

Proving that quotient subgroups of congruent subgroups of modular group are finite.

Published on 21 Dec 2019 - 11:26

score 229 · Answer 2 · 2019-12-26 18:16:40

229

Views

Prove or disprove: For every integer $n$, the quotient group of $\Bbb Z/n\Bbb Z$ modulo any subgroup is cyclic.

Published on 26 Dec 2019 - 18:16

score 468 · Answer 3 · 2019-12-28 02:42:28

468

Views

How to show that the dihedral group $D_{2\cdot 8}$ is the quotient of the free group on $2$ generators by a certain normal subgroup?

Published on 28 Dec 2019 - 2:42

score 144 · Answer 4 · 2019-12-29 23:41:57

144

Views

Is the following quotient group isomorphic to a subgroup?

Published on 29 Dec 2019 - 23:41

score 168 · Answer 5 · 2020-01-02 00:44:35

168

Views

Fulfilling the closure property for the quotient group $G/H$

Published on 02 Jan 2020 - 0:44

score 93 · Answer 6 · 2020-01-03 16:16:27

93

Views

Correspondence theorem: proof explanation

Published on 03 Jan 2020 - 16:16

score 373 · Answer 7 · 2020-01-05 05:21:34

373

Views

$G/H$ is cyclic $\iff$ there is an element $a \in G$ such that $\forall x \in G$, $\exists n \in \mathbb Z$ such that $xa^n \in H$

Published on 05 Jan 2020 - 5:21

score 498 · Answer 8 · 2020-01-07 07:26:16

498

Views

Verifying my proposed proof: If $(G:H)=m$, the order of every element of $G/H$ is a divisor of $m$

Published on 07 Jan 2020 - 7:26

score 90 · Answer 9 · 2026-03-22 19:31:12

90

Views

Recommended Exercise Lists for Quotient Objects

Published on 22 Mar 2026 - 19:31

score 116 · Answer 10 · 2026-03-25 18:14:16

116

Views

Making the module structure trivial

Published on 25 Mar 2026 - 18:14

score 217 · Answer 11 · 2020-01-25 00:14:49

217

Views

Does $A/B \cong C/D$ and $B \cong D$ imply $A \cong C$?

Published on 25 Jan 2020 - 0:14

score 85 · Answer 12 · 2020-01-25 17:53:24

85

Views

Showing that the matrix of the generator has nonzero determinant means the group is finite

Published on 25 Jan 2020 - 17:53

score 791 · Answer 13 · 2020-01-27 14:48:57

791

Views

Let $N$ be a normal subgroup of finite group $G$. Prove that if order of $H$ and order of $G/N$ are relatively prime then $H$ is a subgroup of $N$.

Published on 27 Jan 2020 - 14:48

score 48 · Answer 14 · 2020-01-28 14:12:36

48

Views

Example of a supplemented chief factor of a group

Published on 28 Jan 2020 - 14:12

score 264 · Answer 15 · 2026-03-25 03:02:05

264

Views

Chief factor of a group and what are its types?

Published on 25 Mar 2026 - 3:02

List Question

Proving that quotient subgroups of congruent subgroups of modular group are finite.

Prove or disprove: For every integer $n$, the quotient group of $\Bbb Z/n\Bbb Z$ modulo any subgroup is cyclic.

How to show that the dihedral group $D_{2\cdot 8}$ is the quotient of the free group on $2$ generators by a certain normal subgroup?

Is the following quotient group isomorphic to a subgroup?

Fulfilling the closure property for the quotient group $G/H$

Correspondence theorem: proof explanation

$G/H$ is cyclic $\iff$ there is an element $a \in G$ such that $\forall x \in G$, $\exists n \in \mathbb Z$ such that $xa^n \in H$

Verifying my proposed proof: If $(G:H)=m$, the order of every element of $G/H$ is a divisor of $m$

Recommended Exercise Lists for Quotient Objects

Making the module structure trivial

Does $A/B \cong C/D$ and $B \cong D$ imply $A \cong C$?

Showing that the matrix of the generator has nonzero determinant means the group is finite

Let $N$ be a normal subgroup of finite group $G$. Prove that if order of $H$ and order of $G/N$ are relatively prime then $H$ is a subgroup of $N$.

Example of a supplemented chief factor of a group

Chief factor of a group and what are its types?

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