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 · 2026-03-28 04:35:19

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 28 Mar 2026 - 4:35

score 185 · Answer 2 · 2026-03-20 15:29:51

185

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

Published on 20 Mar 2026 - 15:29

score 29 · Answer 3 · 2026-03-28 04:34:48

29

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

Published on 28 Mar 2026 - 4:34

score 281 · Answer 4 · 2026-03-25 14:37:20

281

Views

Classifying automorphisms using a group presentation.

Published on 25 Mar 2026 - 14:37

score 353 · Answer 5 · 2018-03-04 02:53:17

353

Views

Group Theory: Alternate proof of the 2nd Isomorphism Theorem

Published on 04 Mar 2018 - 2:53

score 359 · Answer 6 · 2026-03-26 09:16:22

359

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

Published on 26 Mar 2026 - 9:16

score 65 · Answer 7 · 2026-03-25 09:32:01

65

Views

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

Published on 25 Mar 2026 - 9:32

score 84 · Answer 8 · 2026-03-25 03:00:32

84

Views

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

Published on 25 Mar 2026 - 3:00

score 356 · Answer 9 · 2026-03-28 11:34:11

356

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 28 Mar 2026 - 11:34

score 673 · Answer 10 · 2026-03-28 11:33:37

673

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 28 Mar 2026 - 11:33

score 115 · Answer 11 · 2026-03-27 10:13:32

115

Views

Isomorphism with operation $a^{log b}$

Published on 27 Mar 2026 - 10:13

score 1.2k · Answer 12 · 2026-03-25 11:16:30

1.2k

Views

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

Published on 25 Mar 2026 - 11:16

score 1.1k · Answer 13 · 2018-03-07 19:33:44

1.1k

Views

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

Published on 07 Mar 2018 - 19:33

score 37 · Answer 14 · 2018-03-09 19:58:51

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 118 · Answer 15 · 2026-03-27 08:47:03

118

Views

Ambiguous notation in isomorphic groups

Published on 27 Mar 2026 - 8:47

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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