Question

Homomorphism and Isomorphism importance

score 254 · Answer 1 · 2026-04-08 23:08:35

254

Views

Homomorphism and Isomorphism importance

Published on 08 Apr 2026 - 23:08

score 64 · Answer 2 · 2022-05-24 17:43:41

64

Views

Proving a different version of the Third Isomorphism Theorem for groups

Published on 24 May 2022 - 17:43

score 163 · Answer 3 · 2026-04-01 01:26:44

163

Views

Why simple factorization of semisimple modules is unique?

Published on 01 Apr 2026 - 1:26

score 216 · Answer 4 · 2026-04-05 17:03:17

216

Views

Prove that the conjugate of $h$ by $g$ is an isomorphism

Published on 05 Apr 2026 - 17:03

score 70 · Answer 5 · 2026-04-05 21:47:27

70

Views

Is there a group isomorphism between finite (abelian) groups where one can prove that the inverse has no analytically closed form?

Published on 05 Apr 2026 - 21:47

score 173 · Answer 6 · 2022-06-06 02:00:56

173

Views

Problem 7.21 Lee, introduction to manifolds.

Published on 06 Jun 2022 - 2:00

score 151 · Answer 7 · 2022-06-06 05:41:16

151

Views

Let $G$ be a group and $\langle a \rangle$ be a unique subgroup of $G$ with order $n$. Then $a$ is in the center of $G$.

Published on 06 Jun 2022 - 5:41

score 96 · Answer 8 · 2026-02-23 15:32:46

96

Views

Homology groups of disjoint sets

Published on 23 Feb 2026 - 15:32

score 51 · Answer 9 · 2022-06-11 08:09:03

51

Views

Normality of the subgroup generated by odd order elements

Published on 11 Jun 2022 - 8:09

score 98 · Answer 10 · 2026-03-29 14:03:10

98

Views

Given $2$ subgroups $N,K < G$ with $N$ normal in $G$ and $N \cap K = \{e\}$ and $G = NK$. Prove that $G/N$ is isomorphic to $K$

Published on 29 Mar 2026 - 14:03

score 150 · Answer 11 · 2022-06-22 22:17:16

150

Views

Prove for all $n \geq 2$ we have $S_n \cong A_n \rtimes \mathbb{Z}_2$

Published on 22 Jun 2022 - 22:17

score 44 · Answer 12 · 2022-06-23 15:49:41

44

Views

Under which group conditions is the homomorphism $g_2 f_1 :H_1 \to G\to H_2$ an isomorphism?

Published on 23 Jun 2022 - 15:49

score 375 · Answer 13 · 2026-03-29 14:03:09

375

Views

Prove $(G_1 \times G_2)/(N_1 \times N_2) \cong G_1/N_1 \times G_2/N_2$.

Published on 29 Mar 2026 - 14:03

score 89 · Answer 14 · 2022-06-26 01:49:26

89

Views

Prove $G \cong H \times K$ when $H,K \lhd G$ and the orders of $H,K$ relatively prime with the product of their orders equaling the order of $G$

Published on 26 Jun 2022 - 1:49

score 48 · Answer 15 · 2026-03-29 14:03:10

48

Views

If $H , K \trianglelefteq F_2$ with $F_2/H\cong F_2/K$ then $H=K$

Published on 29 Mar 2026 - 14:03

List Question

Homomorphism and Isomorphism importance

Proving a different version of the Third Isomorphism Theorem for groups

Why simple factorization of semisimple modules is unique?

Prove that the conjugate of $h$ by $g$ is an isomorphism

Is there a group isomorphism between finite (abelian) groups where one can prove that the inverse has no analytically closed form?

Problem 7.21 Lee, introduction to manifolds.

Let $G$ be a group and $\langle a \rangle$ be a unique subgroup of $G$ with order $n$. Then $a$ is in the center of $G$.

Homology groups of disjoint sets

Normality of the subgroup generated by odd order elements

Given $2$ subgroups $N,K < G$ with $N$ normal in $G$ and $N \cap K = \{e\}$ and $G = NK$. Prove that $G/N$ is isomorphic to $K$

Prove for all $n \geq 2$ we have $S_n \cong A_n \rtimes \mathbb{Z}_2$

Under which group conditions is the homomorphism $g_2 f_1 :H_1 \to G\to H_2$ an isomorphism?

Prove $(G_1 \times G_2)/(N_1 \times N_2) \cong G_1/N_1 \times G_2/N_2$.

Prove $G \cong H \times K$ when $H,K \lhd G$ and the orders of $H,K$ relatively prime with the product of their orders equaling the order of $G$

If $H , K \trianglelefteq F_2$ with $F_2/H\cong F_2/K$ then $H=K$

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