Question

Show that $D_{2^n}/Z(D_{2^n})\cong D_{2^{n-1}}$.

score 43 · Answer 1 · 2026-03-29 18:07:49

43

Views

Show that $D_{2^n}/Z(D_{2^n})\cong D_{2^{n-1}}$.

Published on 29 Mar 2026 - 18:07

score 205 · Answer 2 · 2026-04-11 06:31:37

205

Views

Finding all isomorphisms between two groups

Published on 11 Apr 2026 - 6:31

score 94 · Answer 3 · 2026-04-11 09:20:29

94

Views

A finite abelian subgroup of $(\Bbb{C}^\times)^n$ is iso. to direct product of cyclic group and finite abelian subgroup of $(\Bbb{C}^\times)^{n-1}$

Published on 11 Apr 2026 - 9:20

score 141 · Answer 4 · 2026-04-09 04:24:18

141

Views

A finite group of order $mn$ with $m,n$ relatively prime, together with subgroups of orders $m, n$.

Published on 09 Apr 2026 - 4:24

score 110 · Answer 5 · 2023-01-02 21:06:24

110

Views

What is cardinality of a set of all abelian operations on $\mathbb{R}$ up to an isomorphism?

Published on 02 Jan 2023 - 21:06

score 106 · Answer 6 · 2026-03-25 17:24:29

106

Views

Show that $(\Bbb Z, +)$ and $(\Bbb R, +)$ are not isomorphic groups .

Published on 25 Mar 2026 - 17:24

score 255 · Answer 7 · 2026-03-25 17:24:54

255

Views

Show that $(\Bbb Q, +)$ and $(\Bbb R, +)$ are not isomorphic groups.

Published on 25 Mar 2026 - 17:24

score 175 · Answer 8 · 2026-04-11 05:10:15

175

Views

If there exists a homomorphism from group $G$ onto group $H$ and a second homomorphism from $H$ onto $G$. Does this imply $G$ and $H$ are isomorphic?

Published on 11 Apr 2026 - 5:10

score 247 · Answer 9 · 2026-03-28 12:32:46

247

Views

If $G$ is an infinite group such that $G$ is a homomorphic image of $\Bbb Z$ then prove that $G$ is isomorphic to $\Bbb Z$

Published on 28 Mar 2026 - 12:32

score 82 · Answer 10 · 2026-03-25 17:26:51

82

Views

Let $H$ be a normal of order $6$. If $f :G\to G_1$ be an epimorphism of groups s.t. $H\subset{\rm Ker}(f)$, then show $G_1$ is a hom. image of $G/H$.

Published on 25 Mar 2026 - 17:26

score 423 · Answer 11 · 2026-04-07 11:16:05

423

Views

Find all the subgroups of $\Bbb Z/21\Bbb Z$.

Published on 07 Apr 2026 - 11:16

score 186 · Answer 12 · 2026-04-06 06:18:01

186

Views

Show that $\Bbb Z_{16}$ is not a homomorphic image of $\Bbb Z_4×\Bbb Z_4$.

Published on 06 Apr 2026 - 6:18

score 99 · Answer 13 · 2026-03-28 12:32:11

99

Views

Is there a group with a proper subgroup such that the quotient is isomorphic to itself?

Published on 28 Mar 2026 - 12:32

score 82 · Answer 14 · 2026-04-01 17:51:39

82

Views

Let $\phi:G\to H$ be a surjective hom. of groups. Let $\sigma:H\to G,\phi\sigma=id_H.$ Show $G\cong\ker(\phi)\rtimes H.$

Published on 01 Apr 2026 - 17:51

score 65 · Answer 15 · 2026-04-01 17:48:34

65

Views

Subgroup of matrices is isomorphic to a given semidirect product

Published on 01 Apr 2026 - 17:48

List Question

Show that $D_{2^n}/Z(D_{2^n})\cong D_{2^{n-1}}$.

Finding all isomorphisms between two groups

A finite abelian subgroup of $(\Bbb{C}^\times)^n$ is iso. to direct product of cyclic group and finite abelian subgroup of $(\Bbb{C}^\times)^{n-1}$

A finite group of order $mn$ with $m,n$ relatively prime, together with subgroups of orders $m, n$.

What is cardinality of a set of all abelian operations on $\mathbb{R}$ up to an isomorphism?

Show that $(\Bbb Z, +)$ and $(\Bbb R, +)$ are not isomorphic groups .

Show that $(\Bbb Q, +)$ and $(\Bbb R, +)$ are not isomorphic groups.

If there exists a homomorphism from group $G$ onto group $H$ and a second homomorphism from $H$ onto $G$. Does this imply $G$ and $H$ are isomorphic?

If $G$ is an infinite group such that $G$ is a homomorphic image of $\Bbb Z$ then prove that $G$ is isomorphic to $\Bbb Z$

Let $H$ be a normal of order $6$. If $f :G\to G_1$ be an epimorphism of groups s.t. $H\subset{\rm Ker}(f)$, then show $G_1$ is a hom. image of $G/H$.

Find all the subgroups of $\Bbb Z/21\Bbb Z$.

Show that $\Bbb Z_{16}$ is not a homomorphic image of $\Bbb Z_4×\Bbb Z_4$.

Is there a group with a proper subgroup such that the quotient is isomorphic to itself?

Let $\phi:G\to H$ be a surjective hom. of groups. Let $\sigma:H\to G,\phi\sigma=id_H.$ Show $G\cong\ker(\phi)\rtimes H.$

Subgroup of matrices is isomorphic to a given semidirect product

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