Question

Proof verification: For isomorphism $\phi : G\to H$, show that if $e_{G}\in G$ and $e_{H}\in H$, then $\phi (e_{G})=e_{H}$.

score 1.6k · Answer 1 · 2026-04-12 01:45:41

1.6k

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Proof verification: For isomorphism $\phi : G\to H$, show that if $e_{G}\in G$ and $e_{H}\in H$, then $\phi (e_{G})=e_{H}$.

Published on 12 Apr 2026 - 1:45

score 196 · Answer 2 · 2026-05-12 04:16:19

196

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How to show that $(S\cup\{0\},\ge)$ is order-isomorphic to $(S,\ge)$?

Published on 12 May 2026 - 4:16

score 31 · Answer 3 · 2026-04-12 01:45:40

31

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Proof verification: For isomorphism $\phi: G\to H$, show that $|\phi (a)|=|a|$ for all $a\in G$.

Published on 12 Apr 2026 - 1:45

score 285 · Answer 4 · 2026-05-13 02:18:47

285

Views

Classifying automorphisms using a group presentation.

Published on 13 May 2026 - 2:18

score 356 · Answer 5 · 2026-04-13 21:50:55

356

Views

Group Theory: Alternate proof of the 2nd Isomorphism Theorem

Published on 13 Apr 2026 - 21:50

score 360 · Answer 6 · 2026-05-10 15:00:06

360

Views

How do isomorphisms determine equivalence relations on the class of all groups?

Published on 10 May 2026 - 15:00

score 68 · Answer 7 · 2026-05-10 15:02:24

68

Views

Show that $Aut(G) \leq S_{G}$, the symmetric group.

Published on 10 May 2026 - 15:02

score 89 · Answer 8 · 2026-05-10 20:56:13

89

Views

Where is the error in this proof? (possibly bad surjection assumption)

Published on 10 May 2026 - 20:56

score 358 · Answer 9 · 2026-04-16 02:41:11

358

Views

Prove that for isomorphism $\phi: G\to H$, if $K\leq G$, then $\phi (K)=\left \{ \phi(k):k\in K \right \}\leq H$.

Published on 16 Apr 2026 - 2:41

score 675 · Answer 10 · 2026-04-16 11:00:01

675

Views

For isomorphism $\phi:G\to H$, show that $ab=ba$ for $a,b\in G$ iff $\phi(a)\phi(b)=\phi(b)\phi(a)$.

Published on 16 Apr 2026 - 11:00

score 118 · Answer 11 · 2026-04-13 12:01:45

118

Views

Isomorphism with operation $a^{log b}$

Published on 13 Apr 2026 - 12:01

score 1.2k · Answer 12 · 2026-05-10 13:51:40

1.2k

Views

Prove that if $G$ is a cyclic group of order $n$, then $G$ is isomorphic to $\mathbb{Z}_{n}$.

Published on 10 May 2026 - 13:51

score 1.1k · Answer 13 · 2026-04-13 05:38:13

1.1k

Views

Show that $Z_5 \times Z_7$ and $Z_{35}$ are isomorphic.

Published on 13 Apr 2026 - 5:38

score 40 · Answer 14 · 2026-04-11 17:28:48

How to solve the exercise: "Find out if the groups $(\mathbb{Z}_2\times\mathbb{Z}_2,+)$ and $(\mathbb{Z}_4,+)$ can or not form an isomorphism"?

score 120 · Answer 15 · 2026-04-12 10:36:07

120

Views

Ambiguous notation in isomorphic groups

Published on 12 Apr 2026 - 10:36

List Question

Proof verification: For isomorphism $\phi : G\to H$, show that if $e_{G}\in G$ and $e_{H}\in H$, then $\phi (e_{G})=e_{H}$.

How to show that $(S\cup\{0\},\ge)$ is order-isomorphic to $(S,\ge)$?

Proof verification: For isomorphism $\phi: G\to H$, show that $|\phi (a)|=|a|$ for all $a\in G$.

Classifying automorphisms using a group presentation.

Group Theory: Alternate proof of the 2nd Isomorphism Theorem

How do isomorphisms determine equivalence relations on the class of all groups?

Show that $Aut(G) \leq S_{G}$, the symmetric group.

Where is the error in this proof? (possibly bad surjection assumption)

Prove that for isomorphism $\phi: G\to H$, if $K\leq G$, then $\phi (K)=\left \{ \phi(k):k\in K \right \}\leq H$.

For isomorphism $\phi:G\to H$, show that $ab=ba$ for $a,b\in G$ iff $\phi(a)\phi(b)=\phi(b)\phi(a)$.

Isomorphism with operation $a^{log b}$

Prove that if $G$ is a cyclic group of order $n$, then $G$ is isomorphic to $\mathbb{Z}_{n}$.

Show that $Z_5 \times Z_7$ and $Z_{35}$ are isomorphic.

How to solve the exercise: "Find out if the groups $(\mathbb{Z}_2\times\mathbb{Z}_2,+)$ and $(\mathbb{Z}_4,+)$ can or not form an isomorphism"?

Ambiguous notation in isomorphic groups

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