# 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).

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### Express U(165) as an internal direct product of subgroups in four different ways.

Express U(165) as an internal direct product of subgroups in four different ways. I am trying to solve this problem. What I know is the following: I understand how is Internal Direct Product related ...
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### How to show direct sum of free product of groups is not isomorphic to free product of direct sum of groups?

I guess that $(\mathbb Z \times \mathbb Z) *\mathbb Z$ is not isomorphic to $\mathbb Z \times (\mathbb Z *\mathbb Z)$ since there's no such associative law. What I try: we can write these two groups ...
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### Writing a quotient group as a product of cyclic groups

Let $G = \mathbb{Z}_{12} \times \mathbb{Z}_{12}$, and let $a$ be a generator of $\mathbb{Z}_{12}$. Consider the subgroup $H$ generated by $(a^4, a^6)$. I need to write $G/H$ as a product of cyclic ...
1 vote
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### Generating set cardinality of subgroups of direct products

Given $G \le C_{a_1} \times C_{a_2} \times \ldots \times C_{a_n}$. Is it true that there exists a generating set $S$ that generates $G$ has at most $n$ elements? Indeed, the statement above is correct ...
1 vote
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### A problem on finite abelian group

I recently started studying group theory and there's a problem I encountered in the book Undergraduate Algebra - Serge Lang that I'm currently unable to solve: Let $(A,+)$ be a finite abelian group of ...
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### Existence of a canonical bijection between $G/H \times H/K$ and $G/K$

Suppose $G$ is a group and $K<H<G$. All of the constructions of a bijection $G/H \times H/K \to G/K$ that I saw go more or less as follows. Choose a set of representatives ${g_i}$ for $G/H$, ...
1 vote
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### Maximal quotient group of direct product

In GAP small group library, The group $$[32,2]=\langle a,b,c\mid a^4=b^4=c^2=1, ba=abc, [a,c]=[b,c]=1 \rangle.$$ We say $G/N$ is a maximal quotient group if there exists no quotient group $G/K$ such ...
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1 vote
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### Is the direct sum of Hilbert spaces a 'topological subspace' of the direct product equipped with the product topology?

I am a bit confused about the topological relation between the direct sum and direct product of separable (possibly infinite dimensional) Hilbert spaces $\{H_n\}$. In order to avoid complications ...
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### Show that if $K\triangleleft A\times B$ is non-abelian, then at least one of $K\cap A,K\cap B$ is non-trivial.

Let $A,B$ be groups, $K\triangleleft A\times B$ be a non-abelian normal subgroup. Show that at least one of the intersection $K\cap A$, $K\cap B$ is non-trivial. My attempts: We assume by ...
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### Suppose $G$ and $H$ are two countably infinite abelian groups s.t. every nontrivial element of $G\times H$ has order $7$. Then $G\cong H$.

Suppose $G$ and $H$ are two countably infinite abelian groups such that every nontrivial element of $G \times H$ has order $7$. Then $G$ is isomorphic to $H$. My idea is that each non-trivial element ...
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### Unique factorization of free products of groups satisfying descending chain condition

I am self-studying group theory, and proving Exercise 11.61 of Rotman's An Introduction to the Theory of Groups, on free products: Let $A_1, \ldots, A_n, B_1, \ldots, B_m$ be indecomposable groups ...
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### Does there exist a group $G$, such that $G\otimes H \cong G$ for all finite groups, $H$?

A friend and I, whom only have elementary knowledge of group theory, were playing around some of the natural transformations between group theory and other categories. We probably shouldn't be playing ...
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### Internal weak product of groups (in the chapter of direct product and direct sum of groups)

In the book of algebra by Hungerford in the course of proof to the theorem that if, i) $G = \bigl\langle \bigcup_{i \in I} N_i \bigl\rangle$, where $\{N_i \mid i\in I\}$ is a family of normal ...
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### Generalization of Chinese remainder theorem to non-normal subgroups

I am trying to see if the following generalization holds or if there are counter-examples. Let $H, K<G$ be subgroups of $G$ (not necessarily normal). It is not too hard to prove that \phi: G\to G/...
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### Subgroup as quotient of normal subgroups [closed]

$N \lhd G$ then we get the group $G/N$. Can any subgroup of $G$ be written in the form $G/N$ ? Can we always identify $G/N$ with a subgroup of $G$ and if so is it unique? (like for internal direct ...
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### Example to definition

A direct sum decomposition is $S=\bigoplus_{a \ge 0} S_a$ (d is degree of forms homegeneous polynomial). I try to find and example of this definition, because I need to understand it. Can I take ...
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### Conjugacy of direct product of groups

All groups are finite here. Let $G=\langle a,b \rangle \times H$, $a,b$ commute, $G_{1}=\langle a \rangle \times H_{1}$ and $G_{2}=\langle a \rangle \times H_{2}$ where $H_{1}$ and $H_{2}$ are ...
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### Is the external direct product of O(2) with itself isomorphic to O(2)? (Orthogonal group of 2 x 2 M-matrices)

Is the external direct product of $O(2)$ with itself isomorphic to $O(2)$ i.e. $O(2) \oplus O(2) = O(2)$ I'm guessing they're not, as to prove that is was, I'd have to define an explicit isomorphism ...
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### When an Isomorphism between two algebras is an equivalence?

I'm currently involved in the study of algebraic structures, and there's a concept that seems to appear every so often. Given an Algebra $A=<A,F_i>$ an isomorphism $f$ is a bijective ...
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### Confusion about the direct sum of abelian/nonabelian groups [closed]

I know that there have been many questions on this site about the relationship between the direct product and direct sum of groups. But it seems they don't address the specific issue that I want to ...
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1 vote
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### Confusion about direct limit of vector spaces

Consider the sequence of vector spaces $i_n : \mathbb R^n\to \mathbb R^{n+1}$ given by the inclusions $x\mapsto (x,0)$. We can consider the direct limit of this sequence, call it $\mathbb R^\infty$. ...
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### Prove or give counter example: $G \cong N \times G/N \Rightarrow N$ has a normal complement.
My original question is the following: Given a short exact sequence $1 \to N \xrightarrow{\iota} G \xrightarrow{\pi} Q \to 1$, we have: \begin{equation*} G \cong N \times Q \ \Longrightarrow \ \...