Question

Why is $\Bbb Z/4\Bbb Z\times \Bbb Z/12\Bbb Z\times \Bbb Z/40\Bbb Z$ not isomorphic to $\Bbb Z/8\Bbb Z\times \Bbb Z/10\Bbb Z\times \Bbb Z/24\Bbb Z$?

score 161 · Answer 1 · 2022-08-11 17:42:09

161

Views

Why is $\Bbb Z/4\Bbb Z\times \Bbb Z/12\Bbb Z\times \Bbb Z/40\Bbb Z$ not isomorphic to $\Bbb Z/8\Bbb Z\times \Bbb Z/10\Bbb Z\times \Bbb Z/24\Bbb Z$?

Published on 11 Aug 2022 - 17:42

score 152 · Answer 2 · 2026-03-29 15:33:22

152

Views

If $G_1\cong G_2$, $H_1\triangleleft G_1$, $H_2 \triangleleft G_2$ and $G_1/H_1\cong G_2/H_2$, then is $H_1\cong H_2$?

Published on 29 Mar 2026 - 15:33

score 76 · Answer 3 · 2026-04-05 20:06:58

76

Views

First Isomorphism Theorem: Does each homomorphism has to be surjective? Is it possible to define an homomorphism $\phi:G\to H$ such that $|G|<|H|$?

Published on 05 Apr 2026 - 20:06

score 75 · Answer 4 · 2026-03-29 15:33:25

75

Views

Determination of order of cosets in a factor group of a finite abelian group.

Published on 29 Mar 2026 - 15:33

score 130 · Answer 5 · 2026-03-30 05:15:32

130

Views

Isomorphism between two presentations of 6 order abelian group

Published on 30 Mar 2026 - 5:15

score 53 · Answer 6 · 2026-04-08 21:30:03

53

Views

Show that the set $H=\{f \in S_4 \mid f(4)=4\} \subset S_4$ is isomorphic with $S_3$.

Published on 08 Apr 2026 - 21:30

score 79 · Answer 7 · 2026-03-29 18:32:43

79

Views

Show examples in which $U_1 \cong U_2$ but $G/U_1 \ncong G/U_2$, where $U_1, U_2 \triangleleft G$

Published on 29 Mar 2026 - 18:32

score 3.3k · Answer 8 · 2026-04-09 01:06:25

3.3k

Views

Do nonconstructive proofs of isomorphism exist?

Published on 09 Apr 2026 - 1:06

score 108 · Answer 9 · 2022-08-26 08:55:19

108

Views

If two groups are isomorphic, does this imply that they have the same number of elements with the same order?

Published on 26 Aug 2022 - 8:55

score 504 · Answer 10 · 2022-08-31 06:13:05

504

Views

Prove Graph isomorphism relation is equivalence relation

Published on 31 Aug 2022 - 6:13

score 392 · Answer 11 · 2026-03-29 04:10:57

392

Views

Prove that the group generated by sum of two subgroups is isomorphic with their cartesian product

Published on 29 Mar 2026 - 4:10

score 69 · Answer 12 · 2022-09-01 02:02:55

69

Views

How to interpret "multiplication" in this representation of $GF(2)$?

Published on 01 Sep 2022 - 2:02

score 75 · Answer 13 · 2022-09-01 03:17:49

75

Views

One-to-one between $\cos(x)$ and $\cos^2(x)$ from $(0,\frac {\pi} 2)$?

Published on 01 Sep 2022 - 3:17

score 95 · Answer 14 · 2022-09-03 08:14:01

95

Views

Let $G$ be a finite group, $H$ and $N$ are normal subgroups of $G$ such that $\gcd(|H|,|N|)=1$. Prove $H \times N \cong H \cdot N$.

Published on 03 Sep 2022 - 8:14

score 89 · Answer 15 · 2022-09-04 04:37:47

89

Views

Are two groups equal or isomorphic?

Published on 04 Sep 2022 - 4:37

List Question

Why is $\Bbb Z/4\Bbb Z\times \Bbb Z/12\Bbb Z\times \Bbb Z/40\Bbb Z$ not isomorphic to $\Bbb Z/8\Bbb Z\times \Bbb Z/10\Bbb Z\times \Bbb Z/24\Bbb Z$?

If $G_1\cong G_2$, $H_1\triangleleft G_1$, $H_2 \triangleleft G_2$ and $G_1/H_1\cong G_2/H_2$, then is $H_1\cong H_2$?

First Isomorphism Theorem: Does each homomorphism has to be surjective? Is it possible to define an homomorphism $\phi:G\to H$ such that $|G|<|H|$?

Determination of order of cosets in a factor group of a finite abelian group.

Isomorphism between two presentations of 6 order abelian group

Show that the set $H=\{f \in S_4 \mid f(4)=4\} \subset S_4$ is isomorphic with $S_3$.

Show examples in which $U_1 \cong U_2$ but $G/U_1 \ncong G/U_2$, where $U_1, U_2 \triangleleft G$

Do nonconstructive proofs of isomorphism exist?

If two groups are isomorphic, does this imply that they have the same number of elements with the same order?

Prove Graph isomorphism relation is equivalence relation

Prove that the group generated by sum of two subgroups is isomorphic with their cartesian product

How to interpret "multiplication" in this representation of $GF(2)$?

One-to-one between $\cos(x)$ and $\cos^2(x)$ from $(0,\frac {\pi} 2)$?

Let $G$ be a finite group, $H$ and $N$ are normal subgroups of $G$ such that $\gcd(|H|,|N|)=1$. Prove $H \times N \cong H \cdot N$.

Are two groups equal or isomorphic?

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