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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1answer
29 views

Is Aut(G×H) isomorphic to Aut(G) × Aut(H) [duplicate]

Please suggest me the proof.Am stuck with it.I saw somewhere that it will be true if (o(G),o(H))=1 but why?
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5answers
66 views

Showing $\Bbb Z\times \Bbb Z$ is not a free group. [closed]

I need a little help with prove that $\Bbb Z\times\Bbb Z$ is not a free group by using the universal property, I thought of creating a map between a subset $S$ of $\Bbb Z\times\Bbb Z$ to a group of ...
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3answers
115 views

The only group $G$ with one $A$ and one $B$ as composition factors is $G = A\times B$ (where $A$ and $B$ are non-abelian, finite and simple)

Is it true that if $A$ and $B$ are two non-abelian finite simple groups, then the only finite group $G$ which has one copy of $A$ and one copy of $B$ as composition factors is $G = A \times B$? If not,...
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0answers
12 views

How do you denote functions inherited by a product space?

Given a map $f:A\to B$, define the map $p_n(f):A^n\to B^n$ as $p_n(f)(a_1,...,a_i)=(f(a_1),...,f(a_i))$. Equivalently, you could say given $f$, $p_n(f)$ is such that $\mathrm{proj}_a\circ p_n(f)=f\...
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2answers
33 views

The direct product of quotients is a quotient of the direct product

I asked a question about if we have in general that if $G=G_1\times \cdots \times G_n$ (where $G_i$ are characteristic in $G$ for $i=1,\cdots ,n$), then $${\rm Out}(G)\cong {\rm Out}(G_1)\times\cdots\...
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0answers
28 views

Is the direct product $H \times K$ necessarily a central extension of $H$ by $K$ or $K$ by $H$?

As far as I know, a central + split extension of a group $K$ by a group $H$ is uniquely the direct product $H \times K$. But is the other direction true as well? That is, is any direct product ...
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0answers
14 views

Proving the universal property of the free dendriform algebra

So a dendriform algebra satisfies the relations $$(a<b)<c = a<(b<c) + a < (b>c)$$ $$(a>b)<c = a> (b<c)$$ $$(a<b)>c + (a>b)>c = (a > b) >c$$ Now also ...
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0answers
63 views

What is internal direct sum or internal direct product in Dummit Foote?

I refer to Dummit Foote Chapter 10.3 specifically pages 351,353,354,356 and 357. Does Exercise 10.3.21 on pages 357 (By the way, there's some errata here. Condition (iii) should be $i_1,...,i_k$) ...
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1answer
52 views

Is an infinite direct product or sum of non-trivial modules not finitely generated?

Based upon these two older questions: Show a direct product is not finitely generated. and $R^\mathbb{N}$ is not finitely generated as an $R$-module, I would like to know the answer to the following ...
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2answers
50 views

Cyclic subgroups of maximum possible order of $\Bbb Z_6\times\Bbb Z_{10}\times\Bbb Z_{15}$ of the form $⟨a⟩\times⟨b⟩\times⟨c⟩.$

I was doing problems from Gallian and I found the following one: Find three cyclic subgroups of maximum possible order of $\mathbb Z_6\times \mathbb Z_{10}\times \mathbb Z_{15}$ of the form $\langle ...
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0answers
39 views

Direct product of groups of homomorphisms

This is really a notational thing that is kind of irking me. Consider the dual group of $G=Z_n\times Z_m$, so $G$ is a finite abelian group. I can get an order $n$ homomorphism $\varphi_n$ from $G$ to ...
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38 views

An isomorphism onto the additive group $\mathbb{Z}\times\mathbb{Z}$

Let $I=\{\alpha,\beta\}$ such that $\alpha\ne\beta$. Let $F(I)$ be the free group constructed on $I$ and $\phi_\alpha,\phi_\beta$ be the canonical injections of $\mathbb{Z}$ into $F(I)$. Write $r=\...
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36 views

Does a two-way short exact sequence split?

Suppose we have two short exact sequences, $0\rightarrow B\rightarrow A\rightarrow C \rightarrow 0$ and $0\rightarrow C\rightarrow A\rightarrow B \rightarrow 0$. Is there anything we can conclude ...
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Question about direct factor and direct product.

If $G$ is a group, $G=KK'$, $K=HH'$, where $K,K'$are normal subgroups of $G$ such that $K \cap K'=\left \{e\right \}$ and $H,H'$ are normal subgroups of $K$ such that $H \cap H'=\left \{e\right \}$. ...
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2answers
118 views

Homomorphisms from $\prod_{i\in\mathbb Z}\mathbb Z $ to $\oplus_{i\in\mathbb Z}\mathbb Z$ that fixes $\oplus_{i\in\mathbb Z}\mathbb Z$

I'm trying to verify that $\prod_{i\in\mathbb Z}\mathbb Z $(the direct product of countably many $\mathbb Z$) is not a coproduct in the category of abelian groups. We know that the coproduct object is ...
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1answer
35 views

The Theory of Finite Groups - An introdution - Hans Kurzweil, Bernd Stellmacher.

Assume that $G$ allows a direct decomposition $$G = E_1 × ··· × E_n$$ that is invariant under A, i.e., $E_i^a \in {E_1,...,E_n}$ for all $a ∈ A$ and $i \in {1,...,n}.$ Under the additional hypothesis ...
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0answers
25 views

Relation between group product and quotient group product

Suppose $N_1,\ldots, N_m$ be sequence of normal subgroups of a group $G$. For each $1\leq i\leq m$, write $K_i=\langle\bigcup_{j\ne i} N_j\rangle$. Suppose $\varphi:G\rightarrow G/K_1\times\ldots\...
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1answer
39 views

Let $H_i$ be a subgroup of $G_i$ for $i=1,2,\dots,n.$ Prove that $H_1×\dots × H_n$ is a subgroup of $G_1 ×\dots × G_n.$

First I suuuuuck at proofs. I think I am on the right track but I need some fine tuning. Or if I am totally off let me know. First we show that $H_i$ is nonempty. Note that since $H_i$ is a subgroup ...
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1answer
23 views

A $p$-group action on a cartesian product.

Let $p$ be a prime number. Let $S$ be a group of order $p^r$ and $T$ a set with $m$ elements. Let $$E=\{A\subset S\times T\ |\ |A|=p^r\}.$$ Then, the mapping $\phi:S\rightarrow\mathfrak{S}_{S\times T},...
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2answers
68 views

Prove that the symmetric group $S_n$ has a subgroup isomorphic to $\mathbb{Z}_7 \times \mathbb{Z}_7$ iff $n \ge 14$.

I want to prove that the symmetric group $S_n$ has a subgroup isomorphic to $\mathbb{Z}_7 \times \mathbb{Z}_7$ iff $n \ge 14$. One direction seems clear. |$\mathbb{Z}_7 \times \mathbb{Z}_7| = 49$, ...
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1answer
29 views

Decomposing a group as a direct product of its kernal and image

Suppose $\phi:G\rightarrow H$ is a group homomorphism. When is it true that $G\cong$ ker$(\phi)\oplus G/$ker$(\phi)$? If $G$ and $H$ are abelian and there exists a homomorphism $\varphi:H\...
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1answer
52 views

Direct product is cancellative for finite groups.

Let $A, B, C$ be finite abelian groups such that $A\oplus B\cong A\oplus C$. Then $B\cong C$. $(*)$ Is there an elementary proof for the result $(*)$ above? All the proofs that I have seen thus far ...
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0answers
68 views

Generalization of direct product in group theory

Let $G_1$, $G_2$ and $H$ be groups and $\phi_1: H \to G_1$ and $\phi_2: H \to G_2$ two monomorphisms. Consider the group $(G_1 \times G_2)/N$ where $N$ is the smallest normal subgroup containing the ...
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1answer
57 views

The product of finitely many cyclic groups is cyclic iff the order of the groups are co-primes.

$\mathbb{Z}_m \times \mathbb Z_n$ is cyclic if and only if $\gcd(m,n)=1$. I know the question is a duplicate, I understand the proof for the case of two groups, but for some reason, I can't prove the ...
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the only two irreducible representations of the algebra of $2$-by-$2$ matrices

Let $U$ and $V$ be the only two finite dimensional irreducible representations of a finite dimensional algebra $A$ over a field and let $M$ be the set (algebra) of $2$-by-$2$ matrices over $A$. Is it ...
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2answers
43 views

Direct Product of a Clean Rings

Commutative rings whose elements are a sum of an unit and idempotent by Anderson & Camillo (2002) Definition 1. A commutative ring $R$ is a clean ring if every element $x\in R$ can be written in ...
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1answer
59 views

Direct Product vs. Cartesian Product

I'm currently reading through an introduction to topology book in which the first chapter is an overview of set theory. In this chapter, the Cartesian Product of two sets: $$A \times B $$ is ...
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1answer
48 views

Prove direct product of elements of subgroup is a subgroup of direct products of the group

Let $(G_1,*),(G_2,!)$ be 2 groups. Prove that if $H_1$ is a subgroup of $G_1$ and $H_2$ is a subgroup of $G_2$, then $$H_1\times H_2 = \{(g_1,g_2) \in G_1\times G_2 \mid g_1 \in H_1, g_2 \in H_2\}$$ ...
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0answers
35 views

The direct product $K_m\times K_n$ of two complete graphs.

Let $K_n$ and $K_m$ be two complete graphs of $m$ and $n$ vertices, respectively. If I consider their direct product $K_n\times K_m$ (Also known the Kronecker product), I know about the number of ...
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1answer
59 views

Does the following diagram (2) commute by the axioms for a direct limit?

Suppose $I,J$ represents any finite subsets of the set of natural numbers $N$ and ($∏_{i∈I}A_i,φ_{JI})_{i\subset j\subset N}$ be a directed set. Suppose, also that the the direct limit $(A,\varphi_I)_{...
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2answers
62 views

Let $G$ be a group with center $C$. Let $\phi: G/C\to G$ be a homomorphism with $\phi(gC)\in gC,\forall g\in G$. Prove that $G\cong C\times(G/C)$.

Let $G$ be a group and let $C$ denote the center of $G$. Suppose there exists a group homomorphism $\phi: G/C \longrightarrow G$ with the property that $\phi(gC) \in gC$ for all $g \in G$. Prove that $...
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1answer
57 views

Normal subgroup, internal direct product and isomorphism

Let $N_1, N_2$ be two normal subgroups of the group $G$, with $N_1\times N_2$ denoting the external direct product. We need to show that $G$ is an internal direct Product of $N_1, N_2$ iff $$ \varphi:...
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2answers
248 views

Finding $n$ elements of $\mathbb{Z}_n\times\mathbb{Z}_n$ such that their differences are all different

Let $n\geq 3$ be an integer and consider the group $\mathbb{Z}_n\times\mathbb{Z}_n$ under addition. Question: Does there always exist a choice of $n$ elements $$ (a_1,b_1),\dots,(a_n,b_n)\in\...
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0answers
31 views

Should cycles in direct product group be disjoint?

Suppose $ G_{1}, \dots, G_{d}$ are permutation groups. Given $ G_{i}$ some finite groups for $ i=1,...,d$, we define the direct product group $ G_{1}\times ... \times G_{d} $ as the Cartesian product,...
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0answers
30 views

Group $G$ isomorphic to direct product $G \times G$ [duplicate]

I am looking for a group $G$ with $\vert G \vert > 1$ which is isomorphic to the direct product $G \times G$. I have been looking for such a group among non-finite groups, such as $\mathbb{Z}$ or ...
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0answers
47 views

If G is the internal direct product of the groups H and K then show that G is an X-group (operator group).

Let $G = H \times K$ be an internal direct product of the groups H and K. Let $X = \{x\}$ be a singleton. If $g=hk \in G$ where $h\in H$ and $k \in K$ we define $g^x=h$. [Definition: G is an X-...
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0answers
42 views

A group $G_1$ isomorphic to a subgroup and a quotient group of $G$

Let $G$ be the direct product of two groups $G_1$ and $G_2, G:=G_1 \times G_2$. Then we can see that $G_1 \times 1$ is a normal subgroup of G and $G_1 \times 1$ is isomorphic to $G/(1\times G_2)$. In ...
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2answers
37 views

Generators of a direct product of $\Bbb Z_2$ with $\Bbb Z_4$.

Given two groups : $$\mathbb{Z} _{2} \;( =\mathbb{Z}/2\mathbb{Z})$$ $$\mathbb{Z}_{4} \;( =\mathbb{Z}/4\mathbb{Z}) $$ We define the direct product : $$G =\mathbb{Z} _{2} \times \...
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1answer
26 views

External Direct Product

Let $S_n$ denote the symmetric group on $n$ symbols. The group $S_3\oplus(\Bbb Z/2\Bbb Z)$ is isomorphic to which of the following groups? 1.$\Bbb Z/12\Bbb Z$ 2.$\Bbb Z/6\Bbb Z \oplus \Bbb Z/2\Bbb ...
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1answer
67 views

Every abelian $p$-group is the direct product of cyclic groups.

Theorem $:$ Every abelian $p$-group is the direct product of cyclic groups. I have started reading that proof from this Proof Wiki article. Here I have understood everything before the element $b$ is ...
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1answer
36 views

Find a nonabelian subgroup $T$ of $S_3 \times \Bbb Z_4$ of index $2$, generated by elements $x,y$ such that $|x|=6$, $x^3=y^2$, and $yx=x^{-1}y$.

Find a nonabelian subgroup $T$ of $S_3 \times \Bbb Z_4$ of index 2, generated by elements $x,y$ such that $|x|=6$, $x^3=y^2$, and $yx=x^{-1}y$. I gave it a try but still, don't reach the solution. ...
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1answer
36 views

Find identity and inverses of the group $(\Bbb{Z}_6,+)\times(\Bbb{Z}_5,\cdot)$.

Let $(\Bbb{Z}_6,+)$ and $(\Bbb{Z}_5,\cdot)$ be two groups. Define an operation for $(\Bbb{Z}_6,+)\times(\Bbb{Z}_5,\cdot)$ such that $(\Bbb{Z}_6,+)\times(\Bbb{Z}_5,\cdot)$ is a group. Find the ...
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0answers
15 views

Projective property of cartesian product

Given a cartesian product $W=\Pi_{i\in I} V_i$ of vector spaces $(V_i)_i$ over $\mathbb{K}$. Define the natural projection map as $\pi_k:\Pi (x_i)_i\mapsto x_k$. I then need to show that given $R_i \...
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0answers
82 views

Equivalence of conditions for the restricted sum of a finite family of stable subgroups of a group with operators (Bourbaki)

I am having difficulty following the proof of the following proposition proved by Bourbaki in his Algebra book: Let $G$ be a group with operators in $\Omega$ and $(H_{i})_{i\in I}$ a finite ...
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1answer
33 views

Showing that if $H_1 \unlhd H$ and $G_1 \unlhd G$, then $H_1 \times G_1 \unlhd H \times G $

I have to show that for all $(h_1, g_1) \in H_1 \times G_1 $ and all $(h, g) \in H \times G$ it is the case that $(h,g)(h_1, g_1)(h,g)^{-1} \in H_1 \times G_1$. We know that $(h, g)^{-1} = (h^{-1}, g^...
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2answers
47 views

Universal Property of Products in $A$-modules

I am studying some notes in which the second Universal Property of Products in $A$-modules is defined as following: Given an $A$-modules family $\{M_i\}_{i\in I}$, for all test $A$-module $T$ we ...
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0answers
45 views

What's this product of groups?

Let $G=\Bbb Z[\frac12]/\Bbb Z$ be the (additive) Prufer 2-group and let $\Bbb Z$ be the group of integers. Let $X=G\times \Bbb Z$ be the following product: Let $g_n$ be the subgroups of $G$ in ...
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1answer
64 views

Let $H =\mathbb{Z}/2\mathbb{Z}\times G$ I want to show that H is isomorphic to a subgroup of $G$ and vice-versa

I'm trying to solve a few exercises and I have no clue for this one: Let $G$ be a group formed by $(a_1,a_2,...)$, an infinite series of elements of $\mathbb Z/4\mathbb Z$. Show that $G$ is ...
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1answer
46 views

Find number of subgroups of order $p$ in $\mathbb{Z}_p^n$

Let $p$ be a prime. Find number of subgroups of order $p$ in $\mathbb{Z}_p^n = \mathbb{Z}_p\times\cdots\times\mathbb{Z}_p$, '$n$' times. Consider any element $g\ne 1 \in \mathbb{Z}_p^n$, then order ...
4
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2answers
110 views

Does $G\cong H\times K$ imply $H\unlhd G?$

I want to prove the following exercise. If a group $G$ is the direct product of subgroups $H,K$, then $K$ is isomorphic to $G/H$. To prove this, I think I need first to show $H$ is normal in $G$. ...

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