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).
621
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Group action of direct product group $G \times H$ when groups $G$ and $H$ do not commute
I'm studying direct product groups actions. Usually, for a group $G \times H$ acting on a set $X$ one takes the group action $\cdot_{G \times H}$ to be the natural $(g \times h)(x) = g \cdot_G (h \...
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One sided ideals of a semisimple ring.
Let $A$ be a semisimple ring. I'm wondering whether all ideals of $A$ are two sided. I know that all semisimple rings are both left and right semisimple. And, since $A$ is a semisimple module over ...
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Let $G$ be a group isomorphic to $Z_{n_1} \oplus Z_{n_2} \oplus \cdots\oplus Z_{n_k}$.
Gallian's "Contemporary Abstract Algebra", Chapter 8 Problem 44:
Let $G$ be a group isomorphic to $Z_{n_1} \oplus Z_{n_2} \oplus \cdots \oplus Z_{n_k}$. Let $x$ be the
product of all ...
- 179
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Zappa–Szép product vs. direct product
Let $A$ be an abelian group and $G$ a group. Let $\alpha:G\rightarrow{\rm Bij}(A)$ and $\beta: A\rightarrow{\rm Bij}(G)$ be two group homomorphisms. I have tried to prove that the Zappa–Szép product $...
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Express U(165) as an internal direct product of subgroups in four different ways.
Express U(165) as an internal direct product of subgroups in four different ways.
I am trying to solve this problem.
What I know is the following:
I understand how is Internal Direct Product related ...
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55
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How to show direct sum of free product of groups is not isomorphic to free product of direct sum of groups?
I guess that $(\mathbb Z \times \mathbb Z) *\mathbb Z$ is not isomorphic to $\mathbb Z \times (\mathbb Z *\mathbb Z)$ since there's no such associative law. What I try: we can write these two groups ...
- 415
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Writing a quotient group as a product of cyclic groups
Let $G = \mathbb{Z}_{12} \times \mathbb{Z}_{12}$, and let $a$ be a generator of $\mathbb{Z}_{12}$.
Consider the subgroup $H$ generated by $(a^4, a^6)$. I need to write $G/H$ as a product of cyclic ...
- 319
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44
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Generating set cardinality of subgroups of direct products
Given $G \le C_{a_1} \times C_{a_2} \times \ldots \times C_{a_n}$. Is it true that there exists a generating set $S$ that generates $G$ has at most $n$ elements?
Indeed, the statement above is correct ...
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A problem on finite abelian group
I recently started studying group theory and there's a problem I encountered in the book Undergraduate Algebra - Serge Lang that I'm currently unable to solve:
Let $(A,+)$ be a finite abelian group of ...
- 29
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2
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76
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Existence of a canonical bijection between $G/H \times H/K$ and $G/K$
Suppose $G$ is a group and $K<H<G$. All of the constructions of a bijection $G/H \times H/K \to G/K$ that I saw go more or less as follows.
Choose a set of representatives ${g_i}$ for $G/H$, ...
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72
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Maximal quotient group of direct product
In GAP small group library, The group $$[32,2]=\langle a,b,c\mid a^4=b^4=c^2=1, ba=abc, [a,c]=[b,c]=1 \rangle.$$
We say $G/N$ is a maximal quotient group if there exists no quotient group $G/K$ such ...
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Whether two groups in the form of infinite product are isomorphic
Denote by $Z_n$ the cyclic group of order $n$.
I want to determine whether two groups $G = Z_4 \times \prod_{i=1}^\infty (Z_2 \times Z_2 \times Z_4)$ and $H = (Z_2 \times Z_2) \times \prod_{i=1}^\...
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Clarification needed for notation used in Robinson Abstract algebra text's on External direct product of infinitely many groups.
The following is taken from Derek Robinson's abstract algebra text
Let $\{G_\lambda, \lambda\in \wedge \}$ be a set of groups a restricted choice function for the set is a mapping $f:\wedge \...
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Is the direct sum of Hilbert spaces a 'topological subspace' of the direct product equipped with the product topology?
I am a bit confused about the topological relation between the direct sum and direct product of separable (possibly infinite dimensional) Hilbert spaces $\{H_n\}$.
In order to avoid complications ...
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Show that if $K\triangleleft A\times B$ is non-abelian, then at least one of $K\cap A,K\cap B$ is non-trivial.
Let $A,B$ be groups, $K\triangleleft A\times B$ be a non-abelian normal subgroup. Show that at least one of the intersection $K\cap A$, $K\cap B$ is non-trivial.
My attempts:
We assume by ...
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Suppose $G$ and $H$ are two countably infinite abelian groups s.t. every nontrivial element of $G\times H$ has order $7$. Then $G\cong H$.
Suppose $G$ and $H$ are two countably infinite abelian groups such that every nontrivial element of $G \times H$ has order $7$. Then $G$ is isomorphic to $H$.
My idea is that each non-trivial element ...
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The direct sum of modules is generated by their union
I just started studying modules and stumbled upon the fact that:
the direct sum of a family of modules is the submodule generated by their union:
$$\bigoplus_{i\in I}A_{i}=\langle \bigcup_{i\in I}A_{...
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No automorphisms of order $p^2$
Let $H$ be the group of integers mod p, under addition, where
$p$ is a prime number. Suppose that n is an integer satisfying $1 ≤ n ≤ p$,
and let G be the group $H × H × · · · ×H $($n$ factors). Show ...
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Size of maximal subgroups in the direct product of finite groups
Let $\pi(G)$ be the set of all prime divisors of the order of a finite group $G$. Prove that: if $M$ is a maximal subgroup in $D=G \times G$ then $\pi(M)=\pi(G)$.
My attempt:
1) If $G$ is $p$-group ...
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Let $G_1, G_2, ... , G_t$ be finite groups, and define $G = G_1 × G_2 × \cdots × G_t$. If $G$ is cyclic then $G_1,G_2,...,G_t$ must also be cyclic.
This is what I've done:
First, assume that $G$ is cyclic. Then there exists an element $g$ in $G$ such that $g$ generates $G$. Since $G$ is the direct product of the groups $G_1, G_2,..., G_t$, $g$ ...
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The dihedral group $D_n$ cannot be written as a product of two nontrivial groups
Let $n$ be an odd prime. I want to prove that the dihedral group $D_n$ cannot be written as a product of two nontrivial groups. But it seems problematic, as I completely neglect the properties of ...
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Decomposition of Galois group into direct product using the main theorem
I have a question concerning the main theorem of Galois theory: If $K$ is a field with finite Galois extensions $M,Z$, so that $K\subset M\subset Z$, the theorem says that $$Gal(Z/K)/Gal(Z/M)\cong Gal(...
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100
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Unique factorization of free products of groups satisfying descending chain condition
I am self-studying group theory, and proving Exercise 11.61 of Rotman's An Introduction to the Theory of Groups, on free products:
Let $A_1, \ldots, A_n, B_1, \ldots, B_m$ be indecomposable groups ...
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Exercise 3, Section 6.7 of Hoffman’s Linear Algebra
Let $T$ be a linear operator on a finite-dimensional vector space $V$. Let $R$ be the range of $T$ and let $N$ be the null space of $T$. Prove that $R$ and $N$ are independent if and only if $V=R\...
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Does $G/K \cong H$ imply that $G \cong H\times K$ for normal $H,K$?
I've seen a lot of posts here concerning the reverse statement, but I am wondering whether or not one has the following:
Let $H,K$ be normal subgroups of a group $G$, and assume that $G/K\cong H$. ...
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Confusion regarding proving (by definition) that a group $G$ is a direct sum of $H, K$
For some reason, the distinction between an inner direct sum and an outer direct sum left me a bit confused regarding how one would go about proving that a group $G$ is a direct sum of two other ...
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Find the number of elements of $\Bbb Z_{25}\times \Bbb Z_5$ of order $5.$ [duplicate]
Find the number of elements of $\Bbb Z_{25}\times\Bbb Z_5$ of order $5.$
My solution goes like this:
For $\Bbb Z_{25}\times \Bbb Z_5$, we count $(a,b)\in\Bbb Z_{25}\times\Bbb Z_5$, such that $o((a,b))...
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Is $\mathbb{Z}\oplus \mathbb{Z}_3\cong\mathbb{Z}$ as modules [duplicate]
I'm studying module theory, and I'm trying to prove whether $\mathbb{Z}\oplus \mathbb{Z}_3\cong\mathbb{Z}$ as modules. If i construct the map $\mu :\mathbb{Z}\oplus \mathbb{Z}_3\longrightarrow \mathbb{...
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Show that the $D_{3h}$ prism is given by the direct product $D_{3h}=D_3 \otimes C_s$
I have some questions regarding the notation and solution to the following problem involving the direct product:
The diagram below shows the symmetry operations of an equilateral right triangular ...
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For $K\oplus L$ if $N$ is normal in $L$ then $(K\oplus L) / N = K\oplus (L/N)$
I'm a bit new to direct sums etc., so for the purpose of convincing myself, formally, that $(\mathbb{Z} \oplus \mathbb{Z})/\langle (2,2)\rangle = \mathbb{Z} \oplus \mathbb{Z}_2$, using the basis $\{(1,...
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Does there exist a group $G$, such that $G\otimes H \cong G$ for all finite groups, $H$?
A friend and I, whom only have elementary knowledge of group theory, were playing around some of the natural transformations between group theory and other categories. We probably shouldn't be playing ...
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Internal weak product of groups (in the chapter of direct product and direct sum of groups)
In the book of algebra by Hungerford in the course of proof to the theorem that if,
i) $G = \bigl\langle \bigcup_{i \in I} N_i \bigl\rangle$, where $\{N_i \mid i\in I\}$ is a family of normal ...
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Generalization of Chinese remainder theorem to non-normal subgroups
I am trying to see if the following generalization holds or if there are counter-examples.
Let $H, K<G$ be subgroups of $G$ (not necessarily normal). It is not too hard to prove that
$$\phi: G\to G/...
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Subgroup as quotient of normal subgroups [closed]
$N \lhd G$ then we get the group $G/N$.
Can any subgroup of $G$ be written in the form $G/N$ ?
Can we always identify $G/N$ with a subgroup of $G$ and if so is it unique?
(like for internal direct ...
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Example to definition
A direct sum decomposition is $S=\bigoplus_{a \ge 0} S_a$ (d is degree of forms homegeneous polynomial). I try to find and example of this definition, because I need to understand it.
Can I take ...
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Conjugacy of direct product of groups
All groups are finite here. Let $G=\langle a,b \rangle \times H$, $a,b$ commute, $G_{1}=\langle a \rangle \times H_{1}$ and $G_{2}=\langle a \rangle \times H_{2}$ where $H_{1}$ and $H_{2}$ are ...
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Is the external direct product of O(2) with itself isomorphic to O(2)? (Orthogonal group of 2 x 2 M-matrices)
Is the external direct product of $O(2)$ with itself isomorphic to $O(2)$ i.e.
$O(2) \oplus O(2) = O(2)$
I'm guessing they're not, as to prove that is was, I'd have to define an explicit isomorphism ...
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When an Isomorphism between two algebras is an equivalence?
I'm currently involved in the study of algebraic structures, and there's a concept that seems to appear every so often.
Given an Algebra $ A=<A,F_i> $ an isomorphism $ f $ is a bijective ...
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Confusion about the direct sum of abelian/nonabelian groups [closed]
I know that there have been many questions on this site about the relationship between the direct product and direct sum of groups. But it seems they don't address the specific issue that I want to ...
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Has this 'group product equivalence quotient' construction been substantially studied?
Recently I've seen a few examples of an 'equivalence class' subgroup of a product of two groups: given $G$ and $H$ with homomorphisms $\gamma: G\mapsto K$ and $\eta: H\mapsto K$, one can form a group $...
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If $H,K$ are finite groups with $\text{gcd}(|H|,|K|)=1$, then $\text{Aut}(H \times K) \cong \text{Aut}(H) \times \text{Aut}(K)$.
I have a proof, but it doesn't seem to use the fact that $\text{gcd}(|H|,|K|)=1$, and I can't seem to see where my argument is wrong.
Here is what I did: Define the map $\iota: \text{Aut}(H) \times \...
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Confusion about direct limit of vector spaces
Consider the sequence of vector spaces
$i_n : \mathbb R^n\to \mathbb R^{n+1}$
given by the inclusions $x\mapsto (x,0)$.
We can consider the direct limit of this sequence, call it
$\mathbb R^\infty$.
...
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Prove or give counter example: $G \cong N \times G/N \Rightarrow N$ has a normal complement.
My original question is the following:
Given a short exact sequence $1 \to N \xrightarrow{\iota} G \xrightarrow{\pi} Q \to 1$, we have:
\begin{equation*}
G \cong N \times Q \ \Longrightarrow \ \...
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1
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54
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Neccesary and sufficient condition for a direct product of proper subgroups to be isomorphic to the whole group
It is a known result that if $G$ is a group and $H,K \triangleleft G$ are normal subgroups such that $HK=G$ and $H \cap K= \left\{{1}\right\}$ then $G \cong H \times K$. Is it true that if $H,K \...
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Calculating the direct product and the cross product
I get stuck in the vector calculation from a textbook exercise(A Brief on Tensor Analysis) and hope someone can help me.
The problem is:
show that:
(1)$(\boldsymbol{uv})^T=\boldsymbol{vu}$
(2)for any ...
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169
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How to finish the proof of: If $N\unlhd G=H\times K$, then $N$ is abelian or $N$ intersects $H \times \{e\}$ or $\{e\} \times K$ non trivially
Question: Let $N$ is normal to $G=H\times K$, Prove that $N$ is abelian or $N$ intersects one of $H \times \{e\}$ or $\{e\} \times K$ non trivially.
Solution: Since $G=H\times K$, then $G$ can be ...
- 2,713
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All possible normal subgroups of a direct product. [duplicate]
Let $A$ and $B$ be two groups of co-prime orders. Let $G=A \times B$. Then I want to find all possible normal subgroups of $G$.
It can be shown that it $A'$ and $B'$ are normal subgroups of $A$ and $B$...
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3
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188
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Let $H,K\le G$, abelian $G$, and let $\phi:G\to H$ be a hom. s.t. 1) $\phi(h)=h\forall h\in H$, 2) $\text{Ker }\phi=K$. Show $G=H\oplus K$
Let $H$ and $K$ be subgroups of an Abelian group $G$ and let $\phi:G\rightarrow H$ be a homomorphism such that
$\phi(h)=h$ for all $h\in H$.
$\text{Ker }\phi=K$.
Show that $G=H\oplus K$
Hint: for ...
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Number of subgroups of order $4$ in the direct product $S_2\times S_4$.
I have the following problem:
Find the number of subgroups of 4 elements in the direct product of
permutation groups $S_2 \times S_4$
I started with writing down all the elements in $S_4$ group. ...
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Prove that the group generated by sum of two subgroups is isomorphic with their cartesian product
While studying abstract algebra I encountered the following question:
Given an abelian group $G$ and its two sub-groups $H_1, H_2$, such that $H_1 \cap H_2=\{e\}$, prove that the group generated by $\...
