Question

Part of Birkhoff's HSP theorem for varieties of groups.

score 291 · Answer 1 · 2021-05-04 15:55:32

291

Views

Part of Birkhoff's HSP theorem for varieties of groups.

Published on 04 May 2021 - 15:55

score 110 · Answer 2 · 2021-05-13 15:33:23

110

Views

How do I show a semidirect product of relatively-prime ordered groups is isomorphic to the direct product of those groups?

Published on 13 May 2021 - 15:33

score 172 · Answer 3 · 2021-05-14 05:23:52

172

Views

$\Bbb{Z}_{26} \rtimes_{\alpha} \Bbb{Z}_{5} \cong \Bbb{Z}_{26} \times \Bbb{Z}_{5}$ as groups.

Published on 14 May 2021 - 5:23

score 70 · Answer 4 · 2021-05-22 17:10:18

70

Views

Find an element such that its order equals the cardinality of the group of cartesian product

Published on 22 May 2021 - 17:10

score 47 · Answer 5 · 2021-05-25 08:03:27

47

Views

Prove that a group homomorphism $G \times K \rightarrow H$ is a direct product of group homomorphisms

Published on 25 May 2021 - 8:03

score 81 · Answer 6 · 2021-05-25 17:23:09

81

Views

In $G_1 \times G_2$, $C(H_1 \times H_2) = C(H_1) \times C(H_2)$

Published on 25 May 2021 - 17:23

score 60 · Answer 7 · 2021-06-06 05:58:40

60

Views

Group theory : product of center.

Published on 06 Jun 2021 - 5:58

score 188 · Answer 8 · 2021-06-10 00:01:28

188

Views

If $\displaystyle \bigoplus_{i=1}^{n} \mathbb{Z} \cong \bigoplus_{i=1}^{m} \mathbb{Z}$ as groups, then $n=m.$

Published on 10 Jun 2021 - 0:01

score 40 · Answer 9 · 2021-06-12 16:12:25

40

Views

$H$ and$K$ subgroups of $G$ Prove$\Bigl(H\cap K=\left\{1\right\}\land\left(h\in H,k\in K\Rightarrow hk=kh\right)\Bigr)\Rightarrow HK\approx H\times K$

Published on 12 Jun 2021 - 16:12

score 579 · Answer 10 · 2021-06-12 18:37:54

579

Views

Showing $S_n$ is indecomposable.

Published on 12 Jun 2021 - 18:37

score 82 · Answer 11 · 2021-06-16 16:35:36

82

Views

If $|G_{{\rm ab}}|$ and $|Z(G)|$ are coprime for finite $G$, then $G$ has a unique Remak decomposition.

Published on 16 Jun 2021 - 16:35

score 122 · Answer 12 · 2021-06-19 21:41:27

122

Views

Find $|{\rm Aut}(D_8\times S_3)|$ using a particular result about Remak decompositions.

Published on 19 Jun 2021 - 21:41

score 117 · Answer 13 · 2026-02-23 09:32:13

117

Views

Let $G$ be a finite group. Then $G$ is completely reducible iff $G={\rm Soc}(G)$.

Published on 23 Feb 2026 - 9:32

score 514 · Answer 14 · 2021-06-24 14:46:31

514

Views

What theories are preserved under the product?

Published on 24 Jun 2021 - 14:46

score 63 · Answer 15 · 2026-02-23 05:52:54

63

Views

The commutator of the product of 2 groups

Published on 23 Feb 2026 - 5:52

List Question

Part of Birkhoff's HSP theorem for varieties of groups.

How do I show a semidirect product of relatively-prime ordered groups is isomorphic to the direct product of those groups?

$\Bbb{Z}_{26} \rtimes_{\alpha} \Bbb{Z}_{5} \cong \Bbb{Z}_{26} \times \Bbb{Z}_{5}$ as groups.

Find an element such that its order equals the cardinality of the group of cartesian product

Prove that a group homomorphism $G \times K \rightarrow H$ is a direct product of group homomorphisms

In $G_1 \times G_2$, $C(H_1 \times H_2) = C(H_1) \times C(H_2)$

Group theory : product of center.

If $\displaystyle \bigoplus_{i=1}^{n} \mathbb{Z} \cong \bigoplus_{i=1}^{m} \mathbb{Z}$ as groups, then $n=m.$

$H$ and$K$ subgroups of $G$ Prove$\Bigl(H\cap K=\left\{1\right\}\land\left(h\in H,k\in K\Rightarrow hk=kh\right)\Bigr)\Rightarrow HK\approx H\times K$

Showing $S_n$ is indecomposable.

If $|G_{{\rm ab}}|$ and $|Z(G)|$ are coprime for finite $G$, then $G$ has a unique Remak decomposition.

Find $|{\rm Aut}(D_8\times S_3)|$ using a particular result about Remak decompositions.

Let $G$ be a finite group. Then $G$ is completely reducible iff $G={\rm Soc}(G)$.

What theories are preserved under the product?

The commutator of the product of 2 groups

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