Question

If $G \oplus H$ is isomorphic to a proper subgroup of itself, then must the same be true of one of $G$ and $H$?

score 405 · Answer 1 · 2026-03-26 06:23:12

405

Views

If $G \oplus H$ is isomorphic to a proper subgroup of itself, then must the same be true of one of $G$ and $H$?

Published on 26 Mar 2026 - 6:23

score 1.1k · Answer 2 · 2026-03-28 08:49:30

1.1k

Views

Let $C_n$ be a cyclic group. Show that $C_6 ≅ C_3 × C_2$.

Published on 28 Mar 2026 - 8:49

score 228 · Answer 3 · 2026-03-28 01:27:42

228

Views

Is $PGL(n,\Bbb{R}) \cong SL(n+1, \Bbb{R})$ for even n?

Published on 28 Mar 2026 - 1:27

score 41 · Answer 4 · 2019-09-27 04:18:10

41

Views

Let $1< m,n \in \Bbb N$. The map $g: \Bbb Z_{m} \to \Bbb Z_{mk}$ given by $x \mapsto kx$ is a monomorphism

Published on 27 Sep 2019 - 4:18

score 500 · Answer 5 · 2026-03-25 12:48:50

500

Views

If $f$ is a homomorphism, does it map $p$-groups to $p$-groups?

Published on 25 Mar 2026 - 12:48

score 45 · Answer 6 · 2019-09-28 07:08:11

45

Views

Are these graphs isomorphic ? Why?

Published on 28 Sep 2019 - 7:08

score 1.1k · Answer 7 · 2026-03-29 11:43:28

1.1k

Views

A function $\phi(x) = x^3$ from $\mathbb{R}$ under addition to itself is not an isomorphism

Published on 29 Mar 2026 - 11:43

score 145 · Answer 8 · 2026-03-26 07:59:29

145

Views

If $A$ and $B$ are groups then prove that $A\times B\cong B\times A$

Published on 26 Mar 2026 - 7:59

score 243 · Answer 9 · 2019-09-29 21:34:03

243

Views

Are the groups $(\mathbb Z/12\mathbb Z,+)$ and $(\mathbb Z/13\mathbb Z,\times)$ isomorphic?

Published on 29 Sep 2019 - 21:34

score 397 · Answer 10 · 2026-03-25 19:05:39

397

Views

Is there exists such figure $A$ on plane $E^2$ which isometry group is isomorphic to $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$

Published on 25 Mar 2026 - 19:05

score 280 · Answer 11 · 2014-04-08 14:47:36

280

Views

Construct a group isomorphism $\phi : U(20) \to \mathbb Z_2 \oplus \mathbb Z_4$.

Published on 08 Apr 2014 - 14:47

score 4.2k · Answer 12 · 2014-05-03 19:51:57

4.2k

Views

Why is $\Bbb Z_2 \times \Bbb Z_4$ not isomorphic to $\Bbb Z_8$?

Published on 03 May 2014 - 19:51

score 416 · Answer 13 · 2026-02-23 05:57:21

416

Views

If $G$ is complete, then the holomorph of $G$ is isomorphic to $G\times G$.

Published on 23 Feb 2026 - 5:57

score 23.5k · Answer 14 · 2026-03-26 10:58:31

23.5k

Views

Proof that $S_3$ isomorphic to $D_3$

Published on 26 Mar 2026 - 10:58

score 4.5k · Answer 15 · 2014-10-10 13:44:46

4.5k

Views

Let $G$ be a finite group. Show that $G$ is isomorphic to a subgroup of $S_n$ (symmetric group).

Published on 10 Oct 2014 - 13:44

List Question

If $G \oplus H$ is isomorphic to a proper subgroup of itself, then must the same be true of one of $G$ and $H$?

Let $C_n$ be a cyclic group. Show that $C_6 ≅ C_3 × C_2$.

Is $PGL(n,\Bbb{R}) \cong SL(n+1, \Bbb{R})$ for even n?

Let $1< m,n \in \Bbb N$. The map $g: \Bbb Z_{m} \to \Bbb Z_{mk}$ given by $x \mapsto kx$ is a monomorphism

If $f$ is a homomorphism, does it map $p$-groups to $p$-groups?

Are these graphs isomorphic ? Why?

A function $\phi(x) = x^3$ from $\mathbb{R}$ under addition to itself is not an isomorphism

If $A$ and $B$ are groups then prove that $A\times B\cong B\times A$

Are the groups $(\mathbb Z/12\mathbb Z,+)$ and $(\mathbb Z/13\mathbb Z,\times)$ isomorphic?

Is there exists such figure $A$ on plane $E^2$ which isometry group is isomorphic to $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$

Construct a group isomorphism $\phi : U(20) \to \mathbb Z_2 \oplus \mathbb Z_4$.

Why is $\Bbb Z_2 \times \Bbb Z_4$ not isomorphic to $\Bbb Z_8$?

If $G$ is complete, then the holomorph of $G$ is isomorphic to $G\times G$.

Proof that $S_3$ isomorphic to $D_3$

Let $G$ be a finite group. Show that $G$ is isomorphic to a subgroup of $S_n$ (symmetric group).

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