# Questions tagged [direct-product]

For questions about the direct product of groups, rings, fields or categories. Use (group-theory), (ring-theory), (field-theory) or (category-theory).

653 questions
Filter by
Sorted by
Tagged with
22 views

### Most natural definition of a product of smooth manifolds with a smooth manifold structure.

In this question,I want to ask and clarify(for myself) some points regarding the definition of product manifolds,so that I can appreciate the definition better. The definition of any product structure(...
• 3,619
72 views

### What is the difference between direct sum $\bigoplus_{i\in I} M_i$ and restricted product $\prod'_{i \in I}M_i$ of abelian group?

What is the difference between direct sum and restricted product of abelian group? I recently heard that direct limit of direct product is not a direct sum but a restricted product. Until now, I ...
• 6,351
58 views

### Why Do Additive Categories Need Zero Objects? - Motivation

At the moment, I'm trying to develop intuition behind the derived construction of what is an additive category from the (standard) category definition. (i) It seems natural that a category with the ...
• 917
31 views

• 319
39 views

### For groups $F, G, H$, if $G \not\approx H$, does that mean $F \oplus G \not\approx F \oplus H$? [duplicate]

For context, we were asked to prove in my Abstract Algebra class that $$\mathbb{Z}_3 \oplus \mathbb{Z}_9 \not\approx \mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_3, \tag{1}$$ where $\mathbb{Z}_n$...
• 927
52 views

### Order of an element of the external direct product of 3 or more groups

Prove the following: Let $n$ be a natural number bigger or equal to $2$. Let $G_1,G_2,\ldots,G_n$ be groups. Let $(g_1,g_2,\ldots,g_n)$ be an elemement of $G_1\times G_2\times\cdots\times G_n$. The ...
• 97
121 views

### A problem about the composition factors of group G

There is my problem: Let G be a finite group with composition factors $G_1,G_2,...,G_n$, Assume that $G_i$ not isomorphism $G_j$ for all $i \ne j$. Suppose that for any $\sigma \in S_n$, there exists ...
• 416