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 408 · Answer 1 · 2026-05-11 01:21:17

408

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 11 May 2026 - 1:21

score 1.1k · Answer 2 · 2026-04-16 16:58:24

1.1k

Views

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

Published on 16 Apr 2026 - 16:58

score 234 · Answer 3 · 2026-05-04 02:20:07

234

Views

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

Published on 04 May 2026 - 2:20

score 44 · Answer 4 · 2026-04-16 11:15:09

44

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 16 Apr 2026 - 11:15

score 503 · Answer 5 · 2026-05-11 00:27:17

503

Views

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

Published on 11 May 2026 - 0:27

score 48 · Answer 6 · 2026-04-16 01:17:44

48

Views

Are these graphs isomorphic ? Why?

Published on 16 Apr 2026 - 1:17

score 1.1k · Answer 7 · 2026-04-14 01:07:56

1.1k

Views

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

Published on 14 Apr 2026 - 1:07

score 148 · Answer 8 · 2026-05-11 04:39:17

148

Views

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

Published on 11 May 2026 - 4:39

score 246 · Answer 9 · 2026-04-09 15:59:20

246

Views

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

Published on 09 Apr 2026 - 15:59

score 400 · Answer 10 · 2026-05-10 20:40:30

400

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 10 May 2026 - 20:40

score 283 · Answer 11 · 2026-04-15 00:51:00

283

Views

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

Published on 15 Apr 2026 - 0:51

score 4.2k · Answer 12 · 2026-04-15 17:31:15

4.2k

Views

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

Published on 15 Apr 2026 - 17:31

score 422 · Answer 13 · 2026-05-06 22:57:05

422

Views

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

Published on 06 May 2026 - 22:57

score 23.5k · Answer 14 · 2026-05-10 23:34:24

23.5k

Views

Proof that $S_3$ isomorphic to $D_3$

Published on 10 May 2026 - 23:34

score 4.5k · Answer 15 · 2026-04-16 02:58:27

4.5k

Views

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

Published on 16 Apr 2026 - 2:58

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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