# 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).

264 questions
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### Cartesian product of 2 dimensional

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