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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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Exercise about a family of homomorphisms between abelian groups

It is considered the category $\mathscr{Ab}$ of abelian groups; a family of homomorphisms: $\{ f_i : M_i \rightarrow N_i | i \in I, M_i,N_i \in \mathscr{Ab} \}$ and constructions $\prod_i f_i$, $\...
3
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1answer
36 views

A finite group $G$ and fixed $k\geq 1$ where for every $n\geq 1$, the $n$-direct product $G^n=G\times\dots\times G$ is $k$-generated?

Does exist a finite group $G$ and fixed $k \geq 1$ such that the $n$-direct product $G^n = G \times \dots \times G$ is $k$-generated for every $n \geq 1$? I suspect the answer is no. Does exist a ...
0
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1answer
35 views

Find a subgroup of $\Bbb Z_4\oplus\Bbb Z_2$ not of the form $H\oplus K$ for some $H\le \Bbb Z_4, K\le \Bbb Z_2$.

This is Exercise 8.28 of Gallian's "Contemporary Abstract Algebra". Answers that use only methods from the textbook prior to the exercise are preferred. Here $G_1\oplus G_2$ is the external direct ...
1
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1answer
50 views

A question regarding groups of order $p^2qr$

When considering finite groups $G$ of order, $|G|=p^2qr$, where $p,q,r$ are distinct primes, let $F$ be a Fitting subgroup of $G$. Then $F$ and $G/F$ are both non-trivial and $G/F$ acts faithfully on $...
1
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2answers
57 views

Is there such a thing as direct product of an infinite number of groups?

I know the definition of the direct product $G\times H$ of two random groups $G$ and $H$. It is also clear to me that this can be extended to a product of a finite number of groups $G_1\times \dots ...
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0answers
26 views

Describing the behaviour of subgroups of the direct product of finite cyclic groups based on the gcd of their orders

I am trying to understand the following proof: I understand most of it just fine, except for the last line. I don't really understand why the projection will show us that $\langle xy \rangle \neq \...
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0answers
43 views

Split Sequences. What is the Group?

See page 49 of this book. Proposition 3.22 An extension (17) splits if $N$ is complete. In fact, $G$ is then direct product of $N$ with the centralizer of $N$ in $G$. I believe (17) refers to $$...
4
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1answer
101 views

Does the specific condition on a normal subgroup of a finite group imply that it is a direct factor? v2.0

Suppose $G$ is a finite group, $H \triangleleft G$, such that $\frac{G}{H}$ is simple and $Var(G) = Var(\frac{G}{H})$ (Here $Var(G)$ stands for minimal group variety containing $G$). Does that imply ...
2
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1answer
47 views

Property of elements in $\operatorname{Hom}_\mathbb Z(\prod_{i=1}^{\infty}\mathbb Z, \mathbb Z)$

Does there exist such an element $f\in \operatorname{Hom}_\mathbb Z(\prod_{i=1}^{\infty}\mathbb Z, \mathbb Z)$ satisfying $f(e_k)=1$ for all $k$, where $e_k$ is the element where the $k$ th ...
4
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1answer
84 views

Does the specific condition on a normal subgroup of a finite group imply that it is a direct factor?

Suppose $G$ is a finite group, $H \triangleleft G$, such that $H$ is simple and $Var(H) = Var(G) = Var(\frac{G}{H})$ (Here $Var(G)$ stands for minimal group variety containing $G$). Does that imply ...
0
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0answers
51 views

About definition of category and theorem 8.2 in Hungerford's book

I am new to categories. I think the following two questions are silly... The first question is about definition of category. I feel confused about the definition given in Hungerford's book. In a ...
5
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3answers
176 views

Does there exist a group that is both a free product and a direct product of nontrivial groups?

Do there exist such nontrivial groups $A$, $B$, $C$ and $D$, such that $A \times B \cong C \ast D$? I failed to construct any examples, so I decided to try to prove they do not exist by contradiction....
2
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1answer
102 views

When is $A\rtimes_{\phi_1} B \cong A\rtimes_{\phi_2} B$?

Suppose $1\to K \stackrel{m}{\rightarrow} G \stackrel{f}{\rightarrow} H \to 1$ is short exact sequence of groups. The followings are equivalent: $(1)\ G\cong K \times H;$ $(2)$ The sequence right ...
2
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1answer
69 views

What is the order of the subgroup $\langle 5\rangle \times \langle 3\rangle$ in $Z_{30} \times Z_{12} ?$

What is the order of the subgroup $\langle 5\rangle \times \langle 3\rangle$ in $Z_{30} \times Z_{12} ?$ I think it should be $12$, since $O(5) = 6$ in $Z_{30}$ and $O(3) = 4$ in $Z_{12}$. The order ...
1
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1answer
36 views

Group of order $pqr$

Assume $G$ is a group of order $pqr$, with $p, q, r$ distinct primes. Let $P, Q, R$ be their corresponding Sylow subgroups. In addition, assume $P\subseteq C(G)$ and $R\subseteq N(Q)$ where $C$ and $N$...
1
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1answer
34 views

Property of Sylow $2$-subgroups of $S_n$ and $S_{n+1}$

The question Let $P$ be a Sylow $2$-subgroup of $S_n$ and let $Q$ be a Sylow $2$-subgroup of $S_{n+1}$. Show $Q\cong P\times C_2$ iff $n\equiv 1\pmod{4}$. My attempt The first direction is ...
0
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0answers
32 views

Minimum number of subgroup whose union is $ \mathbb z_4 \times \mathbb z_4$ .

Consider the group $ \mathbb z_4 \times \mathbb z_4$ of order 16 under component wise addition modulo 4. if G is union of $n$ subgroup of order 4 then minimum value of n is a) 7 b) 4 c) 5 d) 6 ...
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3answers
80 views

What are some applications of subdirect product?

I have studied direct products. I know a few applications of direct products, like group isomorphism, etc. What are some applications of sub-direct product of groups?
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2answers
66 views

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
22 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 ...
1
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1answer
39 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
29 views

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
24 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 $Q_8$?
6
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1answer
49 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 $m=\prod_{i=1}^{k} n_{i}$ ...
1
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1answer
27 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 ...
0
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1answer
19 views

$\{\prod_{i=1}^{r} H_{i}\} \bigcap H_{r+1} = \{e\} \Rightarrow H_{i}\bigcap H_{j} = \{e\} \; \forall i\neq j$?

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
544 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
27 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 ...
1
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1answer
80 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'}(...
2
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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 ...
1
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1answer
44 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, ...
0
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1answer
46 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)
0
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1answer
10 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 ...
1
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1answer
67 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
38 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
34 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 $\phi : G \to H$...
3
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2answers
47 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
24 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
53 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
66 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
48 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
54 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 $H\cap K = \{1\},$...
2
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2answers
94 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 ...
1
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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 ...
4
votes
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 $\...
0
votes
0answers
45 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 ...
1
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1answer
95 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 ...
1
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1answer
86 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.