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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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Construct a non trivial homomorphism from $\mathbb{Z}_2\times\mathbb{Z}_4$ to $\mathbb{Z}_8$

I don't know how to solve this problem since the group $\mathbb{Z}_2\times\mathbb{Z}_4$ is not cyclic. I just know that $\mathbb{Z}_2 \times \mathbb{Z}_4= \{ (0,0), (0,1),(1,0),(1,1),(0,2),(1,2),(0,3),...
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0answers
20 views

Subgroups of direct products of free groups

I am reading the following paper of Miller: http://researchers.ms.unimelb.edu.au/ He says that if $G= F_{1} \times F_{2}$ is a direct product of two free groups and $H$ is a subgroup of $G$, then it ...
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1answer
37 views

What does $O_p(G)$ mean in the context of a Fitting subgroup of $G$?

I asked this question as a comment on this old, probably abandoned question about the Fitting subgroup $F(G)$ of a group $G$. It was stated in the question that $F(G)$ is the product of all $O_p(...
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0answers
21 views

Direct Product cancellation in Hopfian rings

A ring $R$ is called to be 'Hopfian' if every ring homomorphism of $R$ onto $R$ is an automorphism of $R$ Question: Given $R$, $S$, $T$, Hopfian rings and $$R \times S \cong T \times S$$ implies ...
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0answers
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Classification of indecomposable abelian groups and direct product

I am having many questions about abelian groups, indecomposable groups and the direct product. Here goes : 1) Are all the indecomposable abelian groups (which can't be written as non trivial direct ...
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1answer
21 views

Group of order greater than 8 doesn't decompose into a direct product and Sylow 2-subgroup isomorphic quaternion group

Is there a group of order greater than 8 that does not decompose into a direct product such that its Sylow 2-subgroup isomorphic quaternion group $Q_8$?
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1answer
48 views

Is the following cyclic group test wrong in WolframAlpha?

Please see here Cyclic group test Isn't this a mistake? Isn't this a cyclic group generated by $(5,1)$ ? thank you!
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0answers
13 views

Proof of $U_{\frac{m}{n_{i}}}(m) \cong U(n_{i})$?

I was studying Internal Direct Product, and although the book doesn't mentions any such theorem anywhere, still I have a strong feeling that this might be true: (1) Let $m=\prod_{i=1}^{k} n_{i}$ ...
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1answer
21 views

Let $G=H_{1} \times H_{2} \dots \times H_{n}$ if and only if $ \; \forall g \in G$, $g= h_{1}h_{2}\ldots h_n$ where $h_{i} \in H_{i}$ is unique.

Let $G=H_{1} \times H_{2} \dots \times H_{n}$ if and only if $ \; > \forall g \in G$, $g= h_{1}h_{2}\ldots h_n$ where $h_{i} \in H_{i}$ is unique. Here, $H_{i}\lhd G \; \forall i$ I have proved ...
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1answer
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$\{\prod_{i=1}^{r} H_{i}\} \bigcap H_{r+1} = \{e\} \Rightarrow H_{i}\bigcap H_{j} = \{e\} \; \forall i\neq j$?

I am studying Internal Direct Products in Group Theory. $\mbox{Let}$ $G$ be a group and $H_{i}\lhd G\; \forall i=1,2\ldots > n$ Suppose $\{\prod_{i=1}^{r} H_{i}\} \bigcap H_{r+1} = \{e\} ...
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4answers
538 views

In the category $\mathbf{Set}$ is “the product of an empty set of sets a one-element set”?

I was reading these notes on Category Theory and it said (paraphrased to add context): Exercise 4: Explain why in $\textbf{Set}$ (the Category of Sets), the product of an empty set of sets is a one-...
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0answers
15 views

Matrices as Outer Direct Sum of Vector Spaces

So studying linear algebra I encountered the outer direct sum and the direct product of collections of vector spaces (which are the same if the collections are finite). When thinking about the uses of ...
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1answer
66 views

About Sylow subgroup and internal direct product.

Let $G$ be a finite group and $P$ a Sylow $p$-subgroup of $G$. Let $Q$ be a subgroup of $P$. If $Z(Q)$ is a Sylow $p$-subgroup of $C_G(Q)$, is $C_G(Q)$ a internal direct product of $Z(Q)$ and $O_{p'}(...
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0answers
45 views

Showing some element is a zero divisor

Let $R=R_1\oplus R_2$, where each $R_i$ is a commutative ring with unity. Let $(S,\eta)$ be a local subring of $R$. Let $\pi_1$ be the projection of $R$ onto $R_1$. It is also given that $\pi_1|_S$ is ...
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1answer
38 views

Subgroups of $G\times H$

Is it true that if $G,H$ are groups, then every subgroup $S$ of $G\times H$ can be written in the form $S\cong S_G\times S_H$, where $S_G$ and $S_H$ are subgroups of $G$ and $H$ respectively? That is, ...
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1answer
45 views

for any integer $n\ge2$, consider the group ${\bf Z}_{p^n}\oplus{\bf Z}_p$. Determine the number of cyclic subgroups of order $p$

I know ${\bf Z}_p\oplus{\bf Z}_p$ has a cyclic group of p+1 order p. but what about when it's n? (p is prime)
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1answer
12 views

Find a nontrivial, proper, normal subgroup of $S_5⊕D_4$ [closed]

I can't find a normal proper nontrivial subgroup for this. I only need it to start a proof.
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1answer
9 views

Basis transformation matrix for direct product space

Suppose $V$ is a n-dimensional linear vector space. $\{s_1, s_2,..., s_n\}$ and $\{e_1, e_2,..., e_n\}$ are two sets of orthonormal basis with basis transformation matrix $U$ such that $e_i = \sum_j ...
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1answer
61 views

Direct product vs categorical product

Is what we call direct products (of groups, vector spaces, modules,...) actually the same as what we call categorical products? Why the word "direct"?
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1answer
30 views

producing element of infinite order in direct product of quotient groups

The question is to produce an element that has infinite order in $\prod_{i} \mathbb{Z}/p_i\mathbb{Z}$ such that $i$ is over $\mathbb{Z}^+$ and $p_i$ denotes the $i^{th}$ integer prime. I am having ...
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3answers
31 views

How to find homomorphism of direct product?

Let $G$ be the direct product $\mathbb{Z}_4 \times \mathbb{Z}_9$ and let $H$ be the direct product $\mathbb{Z}_4 \times\mathbb{Z}_3 \times\mathbb{Z}_3$. Find a non-trivial homomorphism $\phi : G \to H$...
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2answers
44 views

Showing Ring Isomorphism of direct product of $n$ rings

For context I will include the original problem and then what I'm actually asking about. Context: I'm trying to show that the ring $$\mathbb{Z}_{r_1} \times \mathbb{Z}_{r_2} \times \ldots \times \...
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1answer
19 views

Cartesian product of 2 dimensional

Let $R=\{(1,1),(2,2),(3,2),(4,1)\}$. Then how can I calculate $R\times R$?
3
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1answer
52 views

Checking well definedness of bijection from $G/H \times H/K$ to $G/K$

For the sequence of subgroups$$K \leq H\leq G$$ I am trying to create a bijection $\phi:G/H\times H/K \rightarrow G/K$ defined by $$\phi(gH, hK) = (gh)K$$ I can show injectivity since $e\in G/K$ is $K$...
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1answer
44 views

Minimal generating set for product of groups

Let $S_1$ be a minimal generating set for a group $G_1$, $S_2$ be a minimal generating set for a group $G_2$. $T_j:G_j\to G_1\times G_2, j=1,2 $. Then can we say $T(S_1) \cup T(S_2)$ is a "minimal" ...
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1answer
47 views

Isomorphism from $U(st) →U(s)\oplus U(t)$

Let $s,t$ are relatively prime then $U(st) $ is isomorphic to $ U(s)\oplus U(t)$. Define a function from $U(st) →U(s)\oplus U(t)$ by $x\rightarrow (x $mod $s,x$ mod $t)$. I proved this is 1-1 ...
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0answers
44 views

Is it true that if $H,K$ are two normal subgroups of $G$ such that $G=HK$ an $H\cap K = \{1\},$ then $G/H \cong K?$

Determine whether the following is true or false. Note that $\cong$ means group isomorphic. Question: Is it true that if $H,K$ are two normal subgroups of $G$ such that $G=HK$ an $H\cap K = \{1\},$...
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2answers
75 views

Can a Cartesian product of nonelements of Sigma Algebras be in the Product Sigma Algebra?

If $F$ and $G$ are sigma algebras, then $F\times G$, i.e. the set of all Cartesian products of elements of $F$ and elements of $G$, is not a sigma algebra. But it is possible to define a product ...
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1answer
25 views

Redundant condition for internal direct product

My book defines a group $G$ to be the internal direct product of its subgroups $H$ and $K$ if $H$,$K$ are normal in $G$ $H\cap K=${$e$} $G=HK$ From these conditions we can prove that $G≈H×K$. This ...
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1answer
38 views

Characterize Abelian Factor Group to Direct Product

What group is isomorphic to $(\mathbb{Z}_4\times\mathbb{Z}_4\times\mathbb{Z}_8)/\langle(1,2,4)\rangle$? I can only see that $\mathbb{Z}_4\times\mathbb{Z}_4\times\mathbb{Z}_8$ has $128$ elements and $\...
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0answers
42 views

Number of (cyclic) subgroups of $\mathbb{Z}_{16}\times\mathbb{Z}_{30}$

How many subgroups does $\mathbb{Z}_{16}\times\mathbb{Z}_{30}$ have? How many of them are cyclic? First, I would like to know, if this answer for the first part of the question is correct: We ...
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1answer
85 views

Torsion Coefficient in Group Theory

I have seen a calculation about torsion coefficient Determining torsion coefficients But I am now facing another similar question and not sure about is my answer correct. The question: Find the ...
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1answer
61 views

Relationship between Internal direct product and External direct product of groups [closed]

Can somemone explain to me why the internal and external direct products are essentially the same thing? Thank you in advance.
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0answers
36 views

Quotient of finite, nilpotent group by Frattini subgroup is isomorphic to product of quotients of Sylow subgroups by their respective Frattini groups.

Let $G$ be a finite nilpotent group. We know that $G=G_{p_1}\times G_{p_2}\times \cdots \times G_{p_r}$ where $G_{p_i}\in Syl_{p_i}(G)$, $i=1,\dots,r$. Is the following equation right? And why? $...
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0answers
21 views

Dual terminology to “component function”?

Let $V, W_1$ and $W_2$ be vector spaces. A linear map $T : V \to W_1 \times W_2$ is completely determined by its component functions $T_i = \pi_i \circ T$, where $\pi_i$ denotes the standard ...
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1answer
66 views

Isomorphism from inner direct product to external.

Prompt me, please, why in this proof of existence of products' isomorphism the one use, in the last equivalence: $\alpha(ac,bd) = \alpha((a,b),(c,d))$ and for what purpose, in this case, had he used ...
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0answers
55 views

Why does the direct sum of infinitely many groups require only finitely many are nonzero? [duplicate]

Why does the direct sum of infinitely many groups require only finitely many are nonzero? Is it just a case that the direct sum and direct product are two different things, and those are their names? ...
3
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1answer
41 views

How can we describe all maximal proper subgroups of $G \times G$

Suppose G is a finite group, $\mathfrak{M}_G$ is the set of all its maximal proper subgroups. Is there any way to describe $\mathfrak{M}_{G \times G}$ - the set of all maximal proper subgroups of $G \...
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1answer
31 views

Tensorproduct or direct product

In probability theory we had the following proposition about stochastic kernels: Let $P_1$ be a probability measure on $(\Omega_1, \mathcal{A}_1)$, and K a stochastic kernel from $(\Omega_1, \mathcal{...
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1answer
42 views

a nonabelian group $G$ such that it is the internal direct product of $H,K$ with $N$ a normal subgroup in the center trivially intersecting $H,K$

In sequal to my previous question Does it really matter that $G$ must be a nonabelian group in one exercise from Hungerford’s algebra book?, can one find a nonabelian group $G$ that is the internal ...
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0answers
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Does it really matter that $G$ must be a nonabelian group in one exercise from Hungerford’s algebra book?

The following is exercise 7 of chapter one from Hungerford’s algebra book. Let $H,K,N$ be nontrivial normal subgroups of a group $G$ and suppose that $G=H\times K$. Prove that $N$ is in the center of ...
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1answer
63 views

Counting elements of order 6 in $D_{12} \times Z_2$

Actually I know how to count number of elements of particular order from direct product .But In this my counting is not matches with answer so I wanted to know where is my mistake lies . I wanted to ...
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0answers
26 views

Description of Center of group and Normalizer of elements in terms of Subgroup used in Direct product

Let $G=K_1\times K_2\times K_3\times K_4\times ......\times K_n$ and $g\in G$ Then I have to describe Center and $N(g)$ in terms of $K_i$. Where $$N(g)=\{x\in G| gx=xg\}.$$ My attempt: $t\in Z(G)$ ...
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1answer
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Example of Group that is not Internal direct product under weaker condition [duplicate]

I know definition of internal direct which is stronger that is $G$ is internal direct product of normal subgroup $N_i$ where $i$ is indexing upto $n$. Then $G=N_1N_2N_3N_4N_5......N_n$ $...
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1answer
73 views

Equivalence of Internal & External Direct Product

(I know that the first part already exists. But I didn't found the second neither here or in books. I also have some further questions.) Theorem. Let $G $ be a group. If $G$ is the ...
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1answer
49 views

Automorphism of Direct Product $Z_3\times Z_3$

I know form Show $\mid Aut(Z_3 \times Z_3) \mid=48$ that order of Automorphism of $Z_3\times Z_3$ is 48 .And which is same as order of $GL_2(F_3)$.But I wanted to show they are isomorphic .How should ...
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1answer
48 views

Showing that an isomorphism exists between $\Bbb Z$ and a subgroup of $\Bbb Z \times \Bbb Z$.

Say we have a subgroup H of $\Bbb Z \times \Bbb Z$, with $H \cong \Bbb Z$. Lets choose $H:=\{(n,2n)|n \in \Bbb Z\}$ as an example. I'm confused about how to prove the isomorphism. I know that a ...
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0answers
31 views

Writing the symmetry group of a cube without using direct products

The symmetry group of a cube $G$ is commonly cited as being $G \cong S_4 \times \mathbb{Z}_2$. However, the definition of an internal direct product (i.e see this link https://groupprops.subwiki.org/...
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1answer
55 views

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

Suppose $G \cong \mathbb{Z}_5 \rtimes_\alpha \mathbb{Z}_3$ with respect to a homomorphism $\alpha:\mathbb{Z}_3 \to \mathrm{Aut}(\mathbb{Z}_5)$. Show that $\alpha$ is trivial and that $G \cong \mathbb{...
1
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4answers
110 views

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

I'm struggling with this a lot. I think it all boils down to my basic understanding of what the ring $\mathbb Z/2\mathbb Z \times \mathbb Z/5\mathbb Z$ is. As much as I know is that it is a ring of ...