List questions from direct-product

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Is the map $G*H \to G \times H$ injective?

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Is free product of groups always bigger that direct product?

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In GAP, How can I check whether a given group is a direct product?

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386

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$V^*$ is isomorphic to the direct product of copies of $F$ indexed by $A$

Published on 25 Mar 2026 - 6:48

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Prove that $\mathbb{Z}_{5}[x]/(x^2+1)$ is isomorphic to $\mathbb{Z}_{5} \times \mathbb{Z}_{5}$.

Published on 23 Mar 2026 - 7:01

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Subdirect products

Published on 23 Mar 2026 - 10:21

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If $(g,h)\in G\times H$ with $|g|=r$ and $|h|=s$, then $|(g,h)|=\operatorname{lcm}(r,s)$.

Published on 23 Mar 2026 - 0:19

64

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Using Direct Proofs in Discrete Math

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Symmetric Direct Product Distributive?

Published on 22 Mar 2026 - 22:05

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Let $G = \mathbb{Z}_{512} \times \mathbb{Z}_{1729}$ and let $H = <([0],[1])>$. Show that $G/H$ is isomorphic to $\mathbb{Z}_{512}$.

Published on 23 Mar 2026 - 9:05

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Prove that there do not exist non-trivial groups $H$ and $K$ such that $\mathbb{Q}\cong H\times K$.

Published on 25 Mar 2026 - 11:14

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What are the units of $\mathbb Z/2\mathbb Z \times \mathbb Z/5\mathbb Z$

Published on 25 Mar 2026 - 14:26

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Prove that for the semidirect product $\mathbb{Z}_5 \rtimes \mathbb{Z}_3$, the homomorphism $\alpha$ is trivial

Published on 19 Mar 2026 - 20:20

81

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Writing the symmetry group of a cube without using direct products

Published on 24 Mar 2026 - 23:46

List Question

Krull dimension of a direct product of rings

Is the map $G*H \to G \times H$ injective?

Is free product of groups always bigger that direct product?

In GAP, How can I check whether a given group is a direct product?

$V^*$ is isomorphic to the direct product of copies of $F$ indexed by $A$

Prove that $\mathbb{Z}_{5}[x]/(x^2+1)$ is isomorphic to $\mathbb{Z}_{5} \times \mathbb{Z}_{5}$.

Subdirect products

If $(g,h)\in G\times H$ with $|g|=r$ and $|h|=s$, then $|(g,h)|=\operatorname{lcm}(r,s)$.

Using Direct Proofs in Discrete Math

Symmetric Direct Product Distributive?

Let $G = \mathbb{Z}_{512} \times \mathbb{Z}_{1729}$ and let $H = <([0],[1])>$. Show that $G/H$ is isomorphic to $\mathbb{Z}_{512}$.

Prove that there do not exist non-trivial groups $H$ and $K$ such that $\mathbb{Q}\cong H\times K$.

What are the units of $\mathbb Z/2\mathbb Z \times \mathbb Z/5\mathbb Z$

Prove that for the semidirect product $\mathbb{Z}_5 \rtimes \mathbb{Z}_3$, the homomorphism $\alpha$ is trivial

Writing the symmetry group of a cube without using direct products

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