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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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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(...
Kishalay Sarkar's user avatar
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1 answer
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 ...
Poitou-Tate's user avatar
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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 ...
JAG131's user avatar
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Finite Direct Sum in relation to Finite Direct Product - Additive Category

The question is have is admittedly very basic, but I struggle to find posts (on StackExchange) or proofs addressing it. Let $A$ denote some arbitrary additive category. Hence, the direct sum $(X_1\...
JAG131's user avatar
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1 vote
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If a p-group is normal then can we say G is going to be a direct product of P

I recently came across a theorem that states that if G is a finite group and every p-group of G is normal then G is isomorphic to the direct product of its Sylow p subgroups. To prove this we use that ...
Bigalos's user avatar
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-1 votes
2 answers
46 views

External direct sum $U_1\oplus U_2$ isomorphic to $U_1+U_2$ does not necessarily imply that $U_1\cap U_2 = \{0\}$ [closed]

Let $V$ be a vector space (not necessarily finite-dimensional) and let $U_1,U_2\subset V$ be subspaces. If $U_1\cap U_2=\{0\}$, then the surjective linear map $\phi\colon U_1\oplus U_2 \to U_1+U_2$ ...
Apollo13's user avatar
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The free product has the direct product as a factor group. What's the corresponding normal subgroup?

Let $G$ and $H$ be groups. Consider the free product $G * H$ and the direct product $G \times H$. There is a particular way of identifying $G \times H$ as a factor group of $G * H$. Namely, the ...
Dannyu NDos's user avatar
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-1 votes
1 answer
42 views

$R^n$ and $R^{(I)}$ not isomorphic if $I$ is infinite [duplicate]

Let $R$ be a ring and $M$ a free and finitely generated $R$-module. I have to show that there cannot be an isomorphism $M \cong R^{(I)}$ with $I$ infinite. (To note: $R^{(I)} := \{(r_i)_{i \in I} | \...
Minerva's user avatar
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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 $\mathbb{Z}_n$...
Mailbox's user avatar
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3 votes
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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 ...
Fuzzy's user avatar
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2 votes
0 answers
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 ...
ckx's user avatar
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let $A$ and $B$ be rings. Any $A\times B$ module is of the form $M\oplus N$ for some $A$-module $M$ and $B$-module $N$ with $(a,b)\cdot (m,n)=(am,bn)$

Let $A$ and $B$ be two rings, and let $L$ be an $A\times B$-module. Prove that $L$ is of the form $M\oplus N$ for some $A$-module $M$ and $B$-module $N$, with $(a,b)\cdot (m,n) = (am,bn)$. Let $M := \...
Squirrel-Power's user avatar
5 votes
1 answer
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When is the center of group contained in the derived subgroup

Let $N$ be a group. Assume that $N$ is torsion-free, finitely generated and nilpotent. I read somewhere that $$ Z(N) \subset [N,N] \iff N \text{ cannot be written as a direct product of groups } N = A ...
noparadise's user avatar
-1 votes
1 answer
97 views

Semidirect product of groups

In recent days I am studying semidirect product of groups and I have come up with the following question which has already been answered here (From semidirect to direct product of groups), but I can't ...
Priya Sarkar's user avatar
3 votes
2 answers
254 views

Dimension of tensor product vs. dimension of direct product

I am really sorry if this is trivial but I come from Physics and it is really confusing for me to understand what is going on and looking at the answers on both Physics and Math SE sent me through a ...
QFTheorist's user avatar
4 votes
1 answer
155 views

Let $K\lhd G$ be s.t. both $K$ and $G/K$ are simple. Show that either $K$ is the only proper normal subgroup of $G$, or $G \cong K \times (G / K)$.

Sorry about the title, I couldn't fit the whole exercise (Exercise 8.1.6, Nicholson Introduction to Abstract Algebra 4th edition): Let $K \triangleleft G$ be such that both $K$ and $G/K$ are simple. ...
iwjueph94rgytbhr's user avatar
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1 answer
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Is there any External Direct Product which is cyclic but it is the product of two non-cyclic groups? [closed]

Is there any External Direct Product which is cyclic but it is the product of two non-cyclic groups? I know that for an EDP, say $G\times G'$ to be cyclic the $\gcd(o(G),o(G'))=1$. But I am unable to ...
Ayush Kumar Singh's user avatar
1 vote
1 answer
70 views

Assuming that $H \cap K = \{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times K$, why do we need both $H, K\mathrel{\unlhd}G$

The question is the same as here, and I understand the proof in it. But I don't know why do we need both $H, K\mathrel{\unlhd}G$? I think just one of them being a normal subgroup will be enough. W.L.O....
林敬珣's user avatar
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1 answer
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If the direct product of two semilattices exists, what does its Hasse diagram look like in terms of its constituent semilattice Hasse diagrams?

This is likely to be a quick question. Definition: A semilattice $(L,\lor)$ is a commutative, idempotent semigroup. The Hasse diagram $H$ of $L$ is with respect to the order $x\le y$ iff $x\lor y=y$....
Shaun's user avatar
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6 votes
1 answer
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"Asymmetry" in the construction of products and coproducts in concrete categories

Mathematical objects like groups, vector spaces, topological spaces, etc. can be regarded as sets endowed with an additonal structure. The formal setting is that of a concrete category: A concrete ...
Paul Frost's user avatar
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5 votes
1 answer
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"Let $G$ be a finite group with a normal subgroup $N\cong S_3$. Show there is $H\le G$ s.t. $G=N\times H$." Does this $=$ really mean $\cong$?

I'm attempting Problem 60 on this problem set. Let $G$ be a finite group with a normal subgroup $N\cong S_3$. Show that there is a subgroup $H$ of $G$ such that $G = N \times H$. I'm guessing the ...
aleph2's user avatar
  • 984
3 votes
1 answer
92 views

Show that the direct product of structures satisfies a Horn sentence

This is exercise 3.4.16 from Mathematical Logic by Ebbinghaus. Formulas which are derivable in the following calculus are called Horn formulas: Horn formulas without free variables are called Horn ...
iwjueph94rgytbhr's user avatar
1 vote
1 answer
108 views

Subgroups of index $2$ of $D_3 \times D_3$

I know that the group $G=D_3 \times D_3$; with $D_3$ is the Dihedral group of order 6; admits exactly $3$ subgroups of index $2$. We can prove this by using the fact that $G$ and $G/G^2$ have the same ...
ayoub-chess's user avatar
10 votes
1 answer
190 views

When are all normal subgroups of a direct product of finite groups a direct product of normal subgroups?

Let ${G_1}$ and ${G_2}$ be finite groups. When are all normal subgroups of their direct product ${{G_1}\times{G_2}}$ the direct product of normal subgroups in ${G_1}$ and ${G_2}$? So when is it true ...
ElectroSchOOp's user avatar
3 votes
2 answers
165 views

How "exotic" are non-local directly irreducible modular group algebras?

Let $G$ be a finite group and $R = \Bbb F_{p^k} G$ a modular group algebra ($p$ divides $|G|$). I would like to know when $R$ is a directly indecomposable ring. It is quite well-known that $R$ is ...
Amateur_Algebraist's user avatar
1 vote
1 answer
42 views

External direct product realized as an internal direct product. What is the meaning of "realize"? Is my isomorphism correct?

I don't quite understand this sentence: "Every external direct product is naturally realized as an internal direct product." Does "realize" mean that $H\times K$ is equal to $H+K$ (...
niobium's user avatar
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3 votes
1 answer
135 views

Let $G_1, G_2$ be non-trivial groups. If $G_1 \times G_2$ is cyclic, then $G_1 \times G_2$ is finite.

Let $G_1, G_2$ be non-trivial groups. If $G_1 \times G_2$ is cyclic, then $G_1 \times G_2$ is finite. I'm asked about the veracity of this statement and I did the following: suppose BWOC that $G_1 \...
J P's user avatar
  • 343
2 votes
1 answer
160 views

Show that $G = HK$

Let $m, n$ be coprime positive integers. Let $G$ be an abelian group of order $mn$. Let $$ H = \{ g^m : g \in G \} \quad K = \{ g^n : g \in G \} \\ $$ Show that $G = HK \cong H \times K$ So far I've ...
GodelEscher's user avatar
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0 answers
52 views

Direct product of groups are isomorphic, but factor groups are not isomorphic [duplicate]

Question: $A,B,C$ are groups and we know $A\times B\simeq A\times C$. Is $B$ isomorphic to $C$? My work: (1) If $A,B,C$ are finite Abel groups, then this proposition is true, because we just need to ...
QiFeng233's user avatar
0 votes
1 answer
38 views

Internal direct product $\mathrm{H= \phi(N_1) \times \phi(N_2)}$

Let $\mathrm{G=N_1 \times N_2}$ be an internal direct product of groups and let $\phi:G \to H$ be a surjective homorphism. Is it true that we also have the internal direct product $\mathrm{H= \phi(N_1)...
J P's user avatar
  • 343
1 vote
1 answer
99 views

$\mathrm{Hom}_A(\prod_i M_i, N) \cong \prod_i \mathrm{Hom}_A(M_i, N)$ and $\mathrm{Hom}_A (M, \bigoplus_i N_i) \cong \prod_i \mathrm{Hom}_A(M, N_i)$?

I am reading the notes of Pierre Shapira on Algebra and Geometry. He writes the following: Let $A$ be a $k$-algebra, $M \in Mod(A)$. Then $\mathrm{Hom}_A(M, \prod_i N_i) \cong \prod_i \mathrm{Hom}_A(M,...
liv's user avatar
  • 41
1 vote
1 answer
309 views

Group action of direct product group $G \times H$ when groups $G$ and $H$ do not commute

I'm studying direct product groups actions. Usually, for a group $G \times H$ acting on a set $X$ one takes the group action $\cdot_{G \times H}$ to be the natural $(g \times h)(x) = g \cdot_G (h \...
qwer1304's user avatar
0 votes
1 answer
76 views

One sided ideals of a semisimple ring.

Let $A$ be a semisimple ring. I'm wondering whether all ideals of $A$ are two sided. I know that all semisimple rings are both left and right semisimple. And, since $A$ is a semisimple module over ...
Ty Perkins's user avatar
1 vote
2 answers
126 views

Let $G$ be a group isomorphic to $Z_{n_1} \oplus Z_{n_2} \oplus \cdots\oplus Z_{n_k}$.

Gallian's "Contemporary Abstract Algebra", Chapter 8 Problem 44: Let $G$ be a group isomorphic to $Z_{n_1} \oplus Z_{n_2} \oplus \cdots \oplus Z_{n_k}$. Let $x$ be the product of all ...
Daniel's user avatar
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2 votes
0 answers
222 views

Zappa–Szép product vs. direct product

Let $A$ be an abelian group and $G$ a group. Let $\alpha:G\rightarrow{\rm Bij}(A)$ and $\beta: A\rightarrow{\rm Bij}(G)$ be two group homomorphisms. I have tried to prove that the Zappa–Szép product $...
N. SNANOU's user avatar
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0 votes
2 answers
497 views

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 ...
devilcallback_'s user avatar
0 votes
0 answers
63 views

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 ...
iefjkfdhfure's user avatar
1 vote
1 answer
104 views

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 ...
Adam_math's user avatar
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1 vote
0 answers
47 views

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 ...
Quan Hoang's user avatar
1 vote
1 answer
138 views

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 ...
Jenn Inn's user avatar
3 votes
2 answers
168 views

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 $G/H$, ...
MCL's user avatar
  • 542
1 vote
0 answers
83 views

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 ...
Yilan Tan's user avatar
3 votes
0 answers
65 views

Whether two groups in the form of infinite product are isomorphic

Denote by $Z_n$ the cyclic group of order $n$. I want to determine whether two groups $G = Z_4 \times \prod_{i=1}^\infty (Z_2 \times Z_2 \times Z_4)$ and $H = (Z_2 \times Z_2) \times \prod_{i=1}^\...
Alex Lee's user avatar
  • 497
2 votes
1 answer
76 views

Clarification needed for notation used in Robinson Abstract algebra text's on External direct product of infinitely many groups.

The following is taken from Derek Robinson's abstract algebra text Let $\{G_\lambda, \lambda\in \wedge \}$ be a set of groups a restricted choice function for the set is a mapping $f:\wedge \...
Seth's user avatar
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2 votes
0 answers
38 views

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 ...
Keith's user avatar
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4 votes
1 answer
75 views

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 ...
Dreamworld2001's user avatar
4 votes
0 answers
274 views

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 ...
nkh99's user avatar
  • 483
0 votes
1 answer
141 views

The direct sum of modules is generated by their union

I just started studying modules and stumbled upon the fact that: the direct sum of a family of modules is the submodule generated by their union: $$\bigoplus_{i\in I}A_{i}=\langle \bigcup_{i\in I}A_{...
Pingu's user avatar
  • 57
2 votes
1 answer
111 views

No automorphisms of order $p^2$

Let $H$ be the group of integers mod p, under addition, where $p$ is a prime number. Suppose that n is an integer satisfying $1 ≤ n ≤ p$, and let G be the group $H × H × · · · ×H $($n$ factors). Show ...
nkh99's user avatar
  • 483
2 votes
0 answers
71 views

Size of maximal subgroups in the direct product of finite groups

Let $\pi(G)$ be the set of all prime divisors of the order of a finite group $G$. Prove that: if $M$ is a maximal subgroup in $D=G \times G$ then $\pi(M)=\pi(G)$. My attempt: 1) If $G$ is $p$-group ...
Khaled's user avatar
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