Question

Prove that if $\varphi$ is injective then $G \cong \varphi(G)$

score 510 · Answer 1 · 2026-04-03 15:35:11

510

Views

Prove that if $\varphi$ is injective then $G \cong \varphi(G)$

Published on 03 Apr 2026 - 15:35

score 264 · Answer 2 · 2021-01-18 16:53:16

264

Views

Are $\mathbb Z\times \mathbb Z/(4\mathbb Z\times 5 \mathbb Z)\simeq \mathbb Z$ as groups?

Published on 18 Jan 2021 - 16:53

score 266 · Answer 3 · 2021-01-19 17:15:50

266

Views

Is $O(4)\cong SO(4)\times C_2$?

Published on 19 Jan 2021 - 17:15

score 205 · Answer 4 · 2026-03-28 10:27:19

205

Views

Construct a "natural" bijection between homomorphisms with domain $G$ and homomorphisms with a factor group $G/H$ as domain.

Published on 28 Mar 2026 - 10:27

score 46 · Answer 5 · 2021-01-21 14:11:22

46

Views

Let $n$ be prime, for which $k \in \mathbb{N}$ is the map $\phi_k$ a group isomorphism.

Published on 21 Jan 2021 - 14:11

score 315 · Answer 6 · 2026-03-28 10:32:30

315

Views

Is $\mathbb{R/Z}$ isomorphic to $\mathbb{Q/Z}$?

Published on 28 Mar 2026 - 10:32

score 48 · Answer 7 · 2026-03-28 12:13:21

48

Views

Showing that $(\mathbb{C}^{}/\mathbb{R}^{},\cdot)\cong (U/U_2,\cdot)$

Published on 28 Mar 2026 - 12:13

score 50 · Answer 8 · 2021-01-25 08:05:57

50

Views

Finding isomorphism between structures

Published on 25 Jan 2021 - 8:05

score 56 · Answer 9 · 2021-01-30 12:29:08

56

Views

Number of non isomorphic groups of order p(p prime)

Published on 30 Jan 2021 - 12:29

score 103 · Answer 10 · 2021-02-01 15:45:59

103

Views

Number of elements of order $n$: explain why $\Bbb{Z}_4\times\Bbb{Z}_4$ and $\Bbb{Z}_4\times\Bbb{Z}_2\times\Bbb{Z}_2$ are not isomorphic.

Published on 01 Feb 2021 - 15:45

score 43 · Answer 11 · 2026-04-07 01:49:37

43

Views

Quotient group isomorphic to $A_4$

Published on 07 Apr 2026 - 1:49

score 150 · Answer 12 · 2026-03-29 03:36:10

150

Views

Are $D_{20}$ and $\mathbb{Z}_{2} \times \mathbb{Z}_{10}$ isomorphic?

Published on 29 Mar 2026 - 3:36

score 52 · Answer 13 · 2021-02-02 05:04:10

52

Views

Prove that $\forall k$ and for any $g \in G$, the sets $A = \{a \in G : a^k = g\}$, $B = \{b \in H : b ^k = f(g)\}$ have the same number of elements.

Published on 02 Feb 2021 - 5:04

score 317 · Answer 14 · 2021-02-03 18:44:05

317

Views

Show that $H$ is a group, and that it is isomorphic to $G$

Published on 03 Feb 2021 - 18:44

score 111 · Answer 15 · 2021-02-04 03:57:53

111

Views

Show that $H$ is isomorphic to the multiplicative group of positive rational numbers.

Published on 04 Feb 2021 - 3:57

List Question

Prove that if $\varphi$ is injective then $G \cong \varphi(G)$

Are $\mathbb Z\times \mathbb Z/(4\mathbb Z\times 5 \mathbb Z)\simeq \mathbb Z$ as groups?

Is $O(4)\cong SO(4)\times C_2$?

Construct a "natural" bijection between homomorphisms with domain $G$ and homomorphisms with a factor group $G/H$ as domain.

Let $n$ be prime, for which $k \in \mathbb{N}$ is the map $\phi_k$ a group isomorphism.

Is $\mathbb{R/Z}$ isomorphic to $\mathbb{Q/Z}$?

Showing that $(\mathbb{C}^{}/\mathbb{R}^{},\cdot)\cong (U/U_2,\cdot)$

Finding isomorphism between structures

Number of non isomorphic groups of order p(p prime)

Number of elements of order $n$: explain why $\Bbb{Z}_4\times\Bbb{Z}_4$ and $\Bbb{Z}_4\times\Bbb{Z}_2\times\Bbb{Z}_2$ are not isomorphic.

Quotient group isomorphic to $A_4$

Are $D_{20}$ and $\mathbb{Z}_{2} \times \mathbb{Z}_{10}$ isomorphic?

Prove that $\forall k$ and for any $g \in G$, the sets $A = \{a \in G : a^k = g\}$, $B = \{b \in H : b ^k = f(g)\}$ have the same number of elements.

Show that $H$ is a group, and that it is isomorphic to $G$

Show that $H$ is isomorphic to the multiplicative group of positive rational numbers.

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