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 subgroup of $\mathbb{Z_2} \times \mathbb{Z_2}$ where both groups involved in the direct product are not subgroups of $\mathbb{Z_2}$

I am tasked in an exercise to do the following: Construct an example of a subgroup of $\mathbb{Z_2} \times \mathbb{Z_2}$ which is not of the form $K \times J$ for some $K < \mathbb{Z_2}$ and $J <...
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Prove that the direct product of $2$ subgroups is a subgroup

I have an exercise where I am tasked to prove that for $2$ subgroups $K < G$ and $J < H$ of $2$ groups $G,H$ the following is a subgroup: $$K \times J \subset G \times H$$ I believe I have done ...
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Is every infinite group nilpotent iff it is direct product of its sylow p-subgroups?

We know that every finite group is nilpotent iff it is direct product of its sylow p-subgroups.$$$$is this also true for infinite groups?
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Solution check request for the question: If $G=AB$ and $A\cap B=N,$ prove that $G/N\cong A/N\times B/N\\$

Can someone please check my solution to the following question please. Let $A$ and $B$ $N$ be normal subgroups of a group $G$ such that $N\subset A$, $N \subset B.$ If $G=AB$ and $A\cap B=N,$ prove ...
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Prove that $G\cong H\times K$ if and only if there are homomorphisms...

I have some questions for the following following exercise which came from Hungerford's undergraduate Abstract algebra An introduction 3rd edition text in chapter 9, section 1. Let $G$ be an additive ...
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Is the following statement in group theory true?

Let $G$ be a group such that $$G=\prod G_i,$$ where above product is arbitrary. Suppose that there exists a subgroup $H$ of $G$ such that $$\prod G_i=H\prod G_i',$$ where $G_i'$ is derived subgroup of ...
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$H \lhd G$, $\pi:G \to H$ is a group homomorphism with $\pi(h)=h$ show that $G \cong H × G/H$

Let $H$ be a normal subgroup of a group $G$ such that there is a group homomorphism $\pi:G \to H$ with $\pi(h)=h$ for all $h \in H$. Prove that $G$ is isomorphic to $H × G/H$ Don't know how to solve ...
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0 answers
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Can we write a subgroup as a product of subgroups? [duplicate]

Let $G$ be a group such that $$G=\prod G_i,$$ where above product is arbitrary. Suppose that there exists a subgroup $H$ of $G$ such that $$\prod G_i=H\prod G_i',$$ where $G_i'$ is derived subgroup of ...
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What is the difference between the cartesian product and direct sum of vectors?

My notes give the cartesian product of the sets $X_1, . . . , X_n$ as $$X_1 × · · · × X_n = \{(x_1, . . . , x_n) : x_i ∈ X_i for 1 \le i \le n\}$$ I believe we can think of a vector space $V$ where $V=...
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Let $G$ be a finite group and $M$ be a maximal subgroup of $G$. If $G = Z(G)M$, then $M$ is normal in $G$

I need to prove that if $G$ is a group, $M$ is a maximal subgroup of $G$ and $Z(G) \nsubseteq M$,then $M \unlhd G$. Is true that $G = Z(G)M$, right? Is this enough?
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Let $K=\{(g,g):g \in G\}$ where $G$ is a group. There are no subgroups $H_1,H_2$ of $G$ such that $K=H_1\times H_2$

How to formally prove the following: Let $K=\{(g,g):g \text{ is a member of G}\}$ where $G$ is a group There are no sub-groups $H_1,H_2$ of $G$ such that $K=H_1 \times H_2$ If I think about it it's ...
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Given groups $G_1, G_2$, if $K$ is a subgroup of $G_1 \times G_2$, are there $H_1\leq G_1$ and $H_2\leq G_2$ such that $K=H_1\times H_2$? [duplicate]

Duplicate Question Isn't a duplicate, I didn't mention they are finite. Question: Given groups $G_1, G_2$, if $K$ is a subgroup of $G_1 \times G_2$, are there $H_1\leq G_1$ and $H_2\leq G_2$ such ...
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Let $G,G'$ be finite groups of orders $m,n$ respectively. What is the order of $G×G'$? [duplicate]

Let $G,G'$ be finite groups of orders $m,n$ respectively. What is the order of $G×G'$? I have started studying serge lang's undergraduate algebra. This is the question from books group theory ...
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Minimal normal subgroups of Product of simple groups

Known Result: Let $$G= S_1 \times S_2 \times\dots\times S_n,$$ where each $S_i$ are non-abelian simple groups. Then $S_i$'s are the minimal normal subgroup of $G$. (Even $S_i$'s are the only minimal ...
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Normal Subgroup of Direct Product Implies Each Component Normal?

If $N_1 \times N_2 \times ... \times N_p$ is a normal subgroup of $G_1 \times G_2 \times ... \times G_p$, is it true that $N_i \unlhd G_i$ for each value of $i$? Conjugating an arbitrary $(n_1, n_2, .....
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Extending a Property of a p-group by Direct Product

I know that if $P$ is a finite $p$-group, say $\mid P \mid = p^a$ for some prime $p$, and that if $N$ is a non-trivial normal subgroup of $P$ then the center of $P$ intersects non-trivially with $N$ (...
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The cartesian product of an abelian and non abelian group [closed]

If I take the Cartesian product of two groups, with one being abelian and the other being non abelian, Is the product always abelian, always non abelian, or can it be either?
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Definition of Internal Group Direct Product - how does it follow that $(a,b) \circ (c,d) = (a \circ c, b \circ d)$?

In my quest to build up a full understanding of the Internal Direct Product in the context of more-or-less general algebraic structures of $1$ operation, I am struggling with the following. We are ...
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If $G,H$ are finite groups, then $G\times G\cong H \times H$ implies $G \cong H$

Proposition. Let $G,H$ be finite groups (abelian or not). Then the following implication holds: $$G\times G\cong H\times H \Rightarrow G\cong H.$$ In the case of $G,H$ both abelian, one can use the ...
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1 vote
1 answer
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If $S$ is a simple subnormal subgroup of $G$. Prove that if $S$ is nonabelian then $S^G$ is a direct product of simple groups isomorphic to $S$.

Here is the question and my solution. I understood the answer discussed here. My question and the solution is slightly different. Which does not use that $T$ is non abelian. Proof : CLAIM-1: $T^G$ is ...
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2 votes
1 answer
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Factors of Direct Composite of Subsemigroups are in fact Normal Subgroups

Seth Warner's "Modern Algebra" (1965), exercise 13.6. Context: self-study, from a many-years-ago maths degree which did not focus deeply on abstract algebra. If a group $G$ is the direct ...
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3 votes
1 answer
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In the category of fields, is $k\times k=k$ if the direct product is well-defined (in the category of fields)?

This is a follow-up to my previous question on why the direct product $\mathbb{R}\times\mathbb{R}$ is not well-defined, where a proof was given to show that there is no way to construct it without ...
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Why is the direct product $\mathbb{R}\times\mathbb{R}$ undefined when $\mathbb{R}$ is viewed as a field?

Reading through the Wiki article on direct products (https://en.wikipedia.org/wiki/Direct_product), an example is given using the set of real numbers, $\mathbb{R}$, viewed as a set endowed with ...
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If $N_i \cap N_1N_2\ldots N_{i-1}N_{i-2}\ldots N_k=\left<e\right>$, with each $N_j \unlhd G$, $N_i\cap (N_1N_2\ldots N_{i-1})=\left<e\right>.$

Let $N_1, N_2,\ldots N_k$ be normal subgroups of a group $G$ with $G = N_1N_2 \ldots N_k$. Assume for any $1 \leq i \leq k$, $N_i \cap (N_1N_2 \ldots N_{i-1}N_{i+1} \ldots N_{k}) = \langle e \rangle$. ...
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1 vote
1 answer
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Gradings and direct sums notation

I'm studying gradings of rings (specifically polynomial rings) and keep coming across notation looking like this: $$\bigoplus_{\rho\in\mathcal{I}} {D_{\rho}}\mathbb{Z}$$ in this situation ${D_{\rho}}$ ...
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Cohomology ring of disjoint union

It's a well-known fact that $H^*(\bigsqcup_{\alpha}X_{\alpha};R)\xrightarrow{\cong}\prod_{\alpha}H^*(X_{\alpha};R)$ that is also in Hatcher Chapter 3, example 3.13. My attempt by definition is that $H^...
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4 votes
3 answers
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Trying to understand the differences between $\mathbb{Z}_2 * \mathbb{Z}_2$ vs $\mathbb{Z}_2 \times \mathbb{Z}_2$

I’m trying to understand the differences between free products and direct products with an example: $\mathbb{Z}_2 * \mathbb{Z}_2$ vs $\mathbb{Z}_2 \times \mathbb{Z}_2$. If I understand correctly, the ...
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1 answer
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Show that there is no subgroup B of $\mathbb Z ^2$ such that $\mathbb Z ^2 / B \cong \mathbb Z_3 \times \mathbb Z_6 \times \mathbb Z_{21}$

I have the following question in group theory: Show that there is no subgroup $B$ of $\mathbb Z ^2$ such that $\mathbb Z ^2 / B \cong \mathbb Z_3 \times \mathbb Z_6 \times \mathbb Z_{21}$ I don't ...
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1 vote
1 answer
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Two competing definitions of the direct sum of vector spaces

There seem to be two competing definitions of the direct sum of vector spaces. The first one characterises it as the same as the Cartesian product for a finite number of vector spaces, and for an ...
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1 answer
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Show subgroup isomorphic to direct product

$T$ is a minimal generating set for $S_n$. $T=${$(i,i+1),i=1,...,n−1$}. For any fixed $x$ with $1≤x<n$, the set $W =$ $T$\ {$x,x+1 $} does not generate $S_n$. $W$ generates some subgroup of $S_n$. ...
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2 answers
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Does there exist some isomorphism from $Q_8$ to $\Bbb Z_4 \times\Bbb Z_2$?

I am wondering if some direct decomposition exists for quaternion group. I think that I am mixing some things, but let me explain and ask for clarification, tips from your side to let me understand my ...
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0 answers
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A direct product $G=\mathbb{Z}_4\times\mathbb{Z}_4\times\mathbb{Z}_4\times\mathbb{Z}_4$ of the cyclic group $\mathbb{Z}_4$ of order $4$.

Let $G=\mathbb{Z}_4\times\mathbb{Z}_4\times\mathbb{Z}_4\times\mathbb{Z}_4=\mathbb{Z}_4^4$ be a direct product of $4$ copies of the the cyclic group $\mathbb{Z}_4$ of order $4$. Can one regards $G$ as ...
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Two exercises by Robinson on supersolvable groups seem to contradict.

This is concerning (part of) Exercise 5.4.5 and Exercise 5.4.6 of Robinson's, "A Course in the Theory of Groups (Second Edition)". I have done the first one; the second might take me a while....
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If $HK$ is a subgroup but not equal to $G$ (where $H,K<G$), then are $H$ and $K$ normal in $HK$?

I have asked a very similar question already and many have answered it also. If $HK$ is a subgroup of $G$ (where $H$ and $K$ are subgroups of $G$), then are $H$ and $K$ normal in $HK$? But all the ...
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-1 votes
1 answer
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Define a binary operation on {e,(12)}×{e,(123),(132)} so that it becomes isomorphic to $S_3$ [closed]

Since {e,(12)} isomorphic to $Z_2$ and {e,(123),(132)} isomorphic to $Z_3$ and gcd(2,3)=1, so, {e,(12)}×{e,(123),(132)} isomorphic to $Z_6$ which is not isomorphic to $S_3$. But can a binary ...
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Is the given condition necessary for $HK$ to be isomorphic to $H\times K$?

$H \triangleleft G,K \triangleleft G, H \cap K=e$, then, $HK$ is a subgroup of $G$ and $HK$ isomorphic to $H\times K$, where, $\triangleleft$ denotes normal subgroup. If $G$ isomorphic to $H\times K$,...
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3 votes
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For $P$ the set of all primes, ${\rm Hom}(\Bbb Z,\sum_{p\in P}\Bbb Z_p)\not\cong\prod_{p\in P}{\rm Hom}(\Bbb Z,\Bbb Z_p)$

${\rm Hom}(\mathbb Z, \sum_{p\in P}\mathbb Z_p)$ and $\prod_{p\in P}{\rm Hom}(\mathbb Z, \mathbb Z_p)$ are not isomorphic where $P$ is the set of all primes. I was checking the elements of each to see ...
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2 votes
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Under what conditions can we say that $G/N \cong S \implies G \cong N \times S$?

For a recent project (which I have since completed) I needed to derive the automorphism group of the cube graph, and I wanted to do so with some reasonable degree of rigor. I defined a group action of ...
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1 vote
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Count the subgroups of order $p^2$ in $\mathbb{Z}_{p^3}\oplus\mathbb{Z}_{p^2}$

How many subgroups of $\mathbb{Z}_{p^3}\oplus\mathbb{Z}_{p^2}$ are there with order equal to $p^2$? My attempt: Let $H$ be a subgroup of $\mathbb{Z}_{p^3}\oplus\mathbb{Z}_{p^2}$ and $|H|=p^2$, then ...
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1 vote
1 answer
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Addition and Tensor product of Vector spaces for beginners : Concrete example

I am looking for a concrete example for expressions like $$ V_A\otimes V_B = V_C\oplus V_D $$ that shows explicitly what the basis elements actually look like. My attempt was the following, lets take $...
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1 vote
1 answer
50 views

Let $G$ be a group and let $H_1,H_2\unlhd G$ such that $H_1\cap H_2 = \{1\}$ and $H_1H_2=G.$ Prove $G\simeq H_1\times H_2$ [duplicate]

I'm having some struggles with the following exercise: Let $G$ be a group and let $H_1,H_2\unlhd G$ such that $H_1\cap H_2 = \{1\}$ and $H_1H_2=G.$ Prove that $G\simeq H_1\times H_2$ Because $H_1H_2=...
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3 votes
1 answer
161 views

Isomorphisms for Infinite Direct Products of Groups

Here is a question from Section 2.13 of Herstein's "Topics in Algebra" (2nd edition): If $G_{1}$, $G_{2}$, $G_{3}$ are groups, prove that $(G_{1} \times G_{2}) \times G_{3}$ is isomorphic ...
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1 vote
1 answer
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Discovering and describing the homomorphisms returned by the sagemath function direct_product()

The four homomorphisms created by the direct product construction are each an example of a more general construction of homomorphisms involving groups $G$, $H$ and $G\times H$. By using the same ...
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3 votes
1 answer
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External direct product of a family of group homomorphisms

The following is from Hungerford's Algebra book: Theorem 8.10. Let $\{f_i\colon G_i\to H_i\mid i\in I\}$ be a family of homomorphisms of groups and let $f=\prod f_i$ be the map $\prod\limits_{i\in I}...
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3 votes
1 answer
68 views

Quotient of the direct product of cyclic groups

It occurs to me the following is true: $$(\mathbb{Z}_n \times \mathbb{Z}_m) / \mathbb{Z}_k \cong \mathbb{Z}_{n/k} \times \mathbb{Z}_m$$ when $k \mid n$. But I fail to see the way to prove that. The ...
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0 answers
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What is an intuition for subdirect irreducibility?

I am reading "A Course in Universal Algebra" by Burris and H.P. Sankappanavar and they provide this definition: "An algebra A is subdirectly irreducible if for every subdirect embedding ...
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1 vote
1 answer
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Is homotopy group of infinite product of spaces a direct sum or a direct product of groups?

The title pretty much says it all. Do I have $\pi_n(\Pi_{i=1}^\infty X_i)\cong \bigoplus\limits_{i=1}^\infty \pi_n(X_i)$ or $\pi_n(\Pi_{i=1}^\infty X_i)\cong \Pi_{i=1}^\infty \pi_n(X_i)$? From what I ...
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3 votes
2 answers
206 views

How many subgroups does $\mathbb{Z}_{13}\times\mathbb{Z}_{13}$ have?

How many subgroups does $\mathbb{Z}_{13}\times\mathbb{Z}_{13}$ have ? My attempt: Firstly, we observe that the possible orders for an element of this group are $1$ and $13$ only. So, we would need to ...
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0 votes
0 answers
19 views

Count the subgroups of order $25$ in $C_{75} \times C_{10}$ [duplicate]

Count the subgroups of order $25$ in $G=C_{75} \times C_{10}$ Is there a general way to tackle this kind of question? That is, find the number of subgroups of a direct product of cyclic groups?. In ...
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1 vote
1 answer
40 views

Every abelian group of order $9p^2$ where $p\equiv 2\bmod 3$, can be written as the direct product of two cyclic subgroups.

Let $G$ an abelian group of order $9p^2$, where $p$ is an odd prime such that $p\equiv 2\bmod 3$, I have to show that $G$ can be written as the direct product of two cyclic subgroups. In this case, I ...
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