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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34 views

How to show that semi-direct product is isomorphic to a direct product

On page 182 of Dummit and Foote Third Edition the following is stated during their classification of groups of order 30. I have worked through the entirety of the process, however I am confused at the ...
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Confusion about direct sums of $\mathbb{Z}/pq\mathbb{Z}$ where $p, q$ prime

I think I have some kind of fundamental misunderstanding of direct products and direct sums of groups. I'm trying to understand the following statement: For prime numbers $p \neq q$, we have $\mathbb{...
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When is commutative reduced ring a finite direct product of domains?

All rings considered are unital and commutative. Intro Direct products of domains are reduced rings. The opposite is not true but I must admit I have troubles finding counter-examples. It holds that ...
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Does $A \times B \cong A \times C$ imply that $B \cong C$? [duplicate]

Suppose that A, B and C are finite groups, and $\times$ is the direct product, is the implication $A \times B \cong A \times C \Rightarrow B \cong C$ true? I have been told that the reverse ...
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Direct product of Hilbert spaces

I know of direct sum in the category of Hilbert spaces. Is there a concept of direct product too, wherein one can consider arbitrary tuples ? (looks like a no, as in order to talk about convergence ...
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Let $G$ be a group that acts on itself via conjugation. Show that $G \times G \cong G \rtimes G$. [duplicate]

Let $G$ be a group that acts on itself via conjugation. Show that $G \times G \cong G \rtimes G$. My attempt: The action is $\alpha: G\to\operatorname{Aut}(G): g \mapsto (h\mapsto h^g := g^{-1}hg)$ ...
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Understanding an example of a finitely generated projective module which is not free.

Here is the example I know: Consider the ring $R = \mathbb Z_2 \times \mathbb Z_2$ and the submodule $\mathbb Z_2 \times \{0\}.$ it is by construction a direct summand of $R$ but certainly not free....
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Write as the internal direct product of two proper subgroups in every possible way.

Let $|a| = 4$ and $|b| = 2$. Write $\left \langle a \right \rangle \times \left \langle b \right \rangle$ as the internal direct product of two proper subgroups in every possible way. Let $G=\left \...
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1answer
123 views

Elements and Conjugacy Classes of a group

Let $G=(C_{p_1} : C_{3}) \times(C_{p_2} : C_{3})$ where $p_1,p_2\equiv{1}\pmod{3}$. How many elements does the group $G$ have of each order? Furthermore, what is the total number of conjugacy classes?...
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Show that exists an epimorphism $\phi_i: T \rightarrow G_i$ and find $\ker(\phi_i)$

Let $G_1,...,G_n$ be groups and $T:=G_1× \cdots × G_n$. Show that $ \forall i \in \{1, \cdots , n \} $ exists an epimorphism $\phi_i: T \rightarrow G_i$ and find $\ker(\phi_i)$. I'm not sure if what ...
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Let $G$ and $H$ be abelian groups. Show that the product $G\times H$ is also abelian. [duplicate]

Let $G$ and $H$ be abelian groups. Show that the product $G\times H$ is also abelian. I have already proved, that for groups $G$ and $H$ finite groups, then the direct product $G\times H$ is cyclical ...
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51 views

Is there always exists a product over a semigroup?

Let $(S,+)$ a semigroup, is there exists a product over $S$, i.e., a binary operation $\cdot$ over $S$ such that $$ x\cdot(y+z)= x \cdot y + x \cdot z,\, (y+z)\cdot x = y\cdot x + z\cdot x,\, x\cdot(y\...
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Classification of group extension. [duplicate]

Direct product, semidirect product and central extension are different types of group extensions. Is there any other type? If there is, can you please give an example with a small finite group?
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41 views

Let $H, K \lhd G$, then $G = HK \iff G / (H \cap K) \cong G / H \times G / K$

This is a followup of this question So, let $ (G,*) $ be a group. Say $ H,K \lhd G $ If $ G $ is finite, then from the previously asked question we conclude: $ \ G = HK \iff G / (H \cap K) \cong G / H ...
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Extending linear maps on modules.

Let $\{M_i\}_{i\in \mathbb{Z}}$ be an index family of $R$-modules. Then we know that the direct sum $ \bigoplus_{i\in \mathbb{Z}}M_i$ is a submodule of a direct product $ \prod_{i\in \mathbb{Z}}M_i$. ...
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1answer
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On the definition of the direct sum of not necessarily abelian groups

Consider the following definition for the direct sum of groups: Definition. Let $(A_i)_{i\in I}$ be a family of groups. Then we define the direct sum of the family $(A_i)_{i\in I}$ as the subgroup of ...
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Direct and tensor product of group representations

I'm a physicist trying to get some group theory basics from WKT. I'm having some trouble with his definitions of direct and tensor products of representations and representation spaces. He defines ...
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Isomorphism between a product of linear subgroups and a subgroup of a general linear group

This question is as follows: Let $G_1 \subset GL_{n_1}(\mathbb{K})$ and $G_2 \subset GL_{n_2}(\mathbb{K})$ be subgroups. Show that $G_1 \times G_2$ is isomorphic to a subgroup of $GL_{n_1+n_2}(\mathbb{...
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1answer
104 views

How to show that that every finitely generated variety is locally finite

Revised Question: From what I read, an algebra is locally finite if every finitely generated sub-algebra is finite, and a variety is locally finite if every member is locally finite (where a variety ...
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Free groups as free product of infinite cyclic groups

Let $S$ be an arbitrary set (countable or uncountable). It is clear that the free abelian group generated by $S$ is isomorphic to the direct sum $$\bigoplus_{s\in S}\mathbb{Z}.$$ Is the free group ...
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If given two groups $G_1,G_2$ of order $m$ and $n$ respectively then the direct product $G_{1}\times G_{2}$ has a subgroup of order $m$.

I'm trying to understand if this statement is false or true, I have tried understanding by an example. For example if $G_1=(\{{\bar0},\bar1\},+_2),G_3=(\{{\bar0},\bar1,\bar2\},+_3)$ then the direct ...
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Proving that a function is surjective and the equality holds

This is exercise 2.76 (ii) from Rotman's "Introduction to the theory of Groups". I was able to do (i) but I have no idea how to proceed to prove (ii).
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Isomorphism between $\mathbb{Z}_{2} \times \{0\}$ and $\langle (0,2) \rangle$

I'm new to group theory and I'm learning about isomorphisms and quotient groups. I'm stuck trying to show that that $\mathbb{Z}_{2} \times \{0\} \cong \langle (0,2) \rangle$. I understand that two ...
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37 views

Subgroups of index 2 of $\mathbb{Z}_2 \times S_3$

I know that the group $G=\mathbb{Z}_2 \times S_3$ has $3$ subgroups of index $2$ (i.e. of $6$ elements). It is easy to see that $2$ of them are $\{0\} \times S_3$ and $\mathbb{Z}_2 \times A_3$, but I ...
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Endomorphisms of a direct sum of Lie groups

Let $G = \mathbb{T}^n \oplus A = \mathbb{T}^n \times A$, where $\mathbb{T}^n = \left( \mathbb{R} / \mathbb{Z} \right)^n$ and $A$ is a finite discrete abelian group. I'm trying to characterize $\...
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207 views

Why does the empty set not get a relation in a cartesian product?

As far as I understand, when $A=\{1,2\}$ and $B=\{1,2\}$, then $A \times B =\{(1,1),(1,2),(2,1),(2,2)\}$. But $\emptyset \in A$ and $\emptyset \in B$. Are any of these valid? If not, why not? a. $(\...
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For what $m,n$ is $D_{mn}$ isomorphic to $D_m \times Z_n$?

For what $m,n$ is $D_{mn}$ isomorphic to $D_m \times Z_n$? Should I try to define an isomorphism between them? Where to start?
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How $ \mathbb {Z}_2 \oplus \mathbb {Z}_2 \oplus \mathbb {Z}_3 \oplus \mathbb {Z}_5 \approx \mathbb {Z}_2 \oplus \mathbb {Z}_6 \oplus \mathbb {Z}_5$

In the book Contemporary abstract algebra by Joseph Gallian in page 166 the following things are said : By using results above in an iterative fashion, one can express the same group (up to ...
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Automorphism group of direct product of groups, proof [duplicate]

I need a proof of the following fact: if a group $G=M \times N$ is the direct product of subgroups $M$ and $N$ such that $\lvert M \rvert$ and $\lvert N \rvert$ are relatively prime, then $\mathrm{Aut}...
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Reference for irreducible representations of direct product of compact lie groups.

In some unpublished notes which are not publically available, a version of the following result is proved: If $\phi_1$ and $\phi_2$ are complex, irreducible representations of compact Lie groups $G_1$...
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1answer
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Can a direct product $R\times X$ of two integral domains $R$ and $X$ ever be an integral domain?

Assuming we are using the standard definition of a direct product, I think this is never the case. An integral domain is a commutative ring with unity that has no zero divisors. Just the fact that ...
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Fundamental theorem of finite Abelian group

In Gallian's Contemporary Abstract Algebra, he mentioned about this Greedy Algorithm for an Abelian group. It's basically a procedure of expressing a finite Abelian group as an internal direct product ...
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2answers
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Show that the group ordering in the direct product is doesn't matter to the group structure.

Show that the group ordering in the direct product is doesn't matter to the group structure. For instance, for $A,B,C \ne \emptyset$, $A×B×C$ and $A×C×B$ is same (That is, the group structure of $A×B×...
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Automorphism of direct product of rings

Let $$R=A_1\times A_2\times\dots\times A_n.$$ Let's assume there are no non trivial automorphisms of these $A_i$s. I claim that the only possible automorphisms of this structure are the permutations ...
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Finite Direct product of solvable groups is solvable

I found this proof. But among my colleagues it is rumored that there is a mistake. Could you tell me if I did, what would the mistake be? I really don't have much idea how this proof works.
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$H$ and $K$ are two normal subgroups of group $G$, $H\cap K=\{1\}$, $G=HK$, how to prove $G$ is isomorphic to $H\times K$

If group $G$ has two normal subgroups $H$ and $K$ satisfying $H\cap K=\{1\}$ and $G=HK$, how to prove that $G$ is isomorphic to $H\times K$? Since the factoring of any $g\in G$ as product $hk$ is ...
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Direct product and commutativity of factors

If $H$ and $K$ are both normal subgroups of $G$, and if every element of $G$ can be written uniquely as $hk$ with $h\in H$ and $k\in K$ (so $G$ is the direct product of $H$ and $K$), does it follow ...
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Associating Projections with Vector Space (Infinite) Internal Direct Sum

If I start from a category theory approach I can define vector space direct sum with inclusions and direct product with projections by the universal properties in each case. Defining the external ...
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Is an infinite direct product of a locally finite group with itself locally finite? [closed]

To be locally finite means that any finitely generated subgroup is finite. If the assertion is untrue, is an infinite direct product of a finite group with itself locally finite?
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Greedy Algorithm for an Abelian Group

In Gallian's Contemporary Abstract Algebra, he mentioned about this Greedy Algorithm for an Abelian group. It's basically a procedure of expressing a finite Abelian group as an internal direct product....
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Is every vector space isomorphic to a direct product of one-dimensional vector spaces?

Question: Is every vector space isomorphic to a direct product of one-dimensional vector spaces? I know every vector space is a direct sum of one-dimensional vector spaces, since every vector space ...
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1answer
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Is the large Sigma notation (summation notation) used for coproducts?

Binary products in a category are denoted $A_1 \times A_2$, while arbitrary products are denoted $\prod_{i \in I} A_i$. Binary products are (sometimes) denoted $A_1 + A_2$,* so I expected that ...
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1answer
67 views

Finding a contradiction to $G$ being abelian when $G\cong M\times L$

I am trying to disprove/prove the following: "Let $M, L \subseteq G$, $G \cong M \times L$. Then if all the elements of $G$ commute with all the elements of $L$, we have that $G$ is abelian."...
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If $G=H\times K$ then can we view $G/K$ as $H$? [duplicate]

Here $K$ is identified as $\{1\}\times K$. I think because $G/K$ does not depend on $H$, it is natural to say that $G/K$ and $H$ is isomorphic.
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3answers
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Finite cyclic groups are isomorphic to their product with $\Bbb Z$?

I'm currently making a start on group theory and have hit a roadblock with a relatively basic theorem on finite cyclic groups. The specific relation killing me is: $$\mathbb{Z}_n \times \mathbb{Z}_m \...
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Fitting group of a direct product

Given the group $G=N\oplus M$, show $F(G)=F(N)\oplus F(M)$, where $F(G)$ denotes the Fitting group of $G$ (the product of all nilpotent normal subgroups). I would say the inclusion $\supseteq$ is ...
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Computation of a quotient group in the direct sum of three copies of Z

My question is simple: How to compute the quotient group $$\frac{\langle(1,1,0),(0,0,1)\rangle}{\langle(1,1,-1),(1,1,1)\rangle}$$ of subgroups of $\mathbb Z \oplus \mathbb Z \oplus \mathbb Z$? The ...
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1answer
47 views

Isomorphism of direct products

If $G, A, B$ are groups, it is not true in general that $G\times A\cong G\times B \implies A\cong B$. For instance, $G=\mathbb{Z}\times\mathbb{Z}\times\cdots, A=\mathbb{Z}, B=\{1\}$. What happens when ...
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1answer
60 views

Find the order of each of the elements in the direct product of $Z_3 \times Z_6$

I am stuck in this homework question. Please help me to find the order of each of the elements in the direct product of $Z_3 \times Z_6$. This is what I got so far. elements of $Z_3 = \{0,1,2\}$ ...
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1answer
60 views

When is $G \subset S_{2n}$ isomorphic to $S_m \times \Bbb Z/2\Bbb Z$?

Consider the subgroup $G \subset S_{2n}$ consisting of permutations $\sigma\in S_{2n}$ such that $\sigma (i) + \sigma (2n+1-i) = 2n+1$. I want to know if it is true that $G \subset S_{2n}$ is ...

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