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Let $A,B$ be groups.Can you explain why $U\le A \times B$ does not imply $U=\left(A\cap U\right) \times \left( B \cap U \right)$ this is an exercise in the book of the theory of finite groups an introuduction written by H.Kurzweil. the meaning of each symbol may as follows. $A\times B$ is the direct product of $A$,$B$. $U$ is the subgroup of $A\times B$ , thus $U= \lbrace\left(a,b\right)|a \in A,b\in B \rbrace$, $A \cap U=\lbrace a_1|\left(a_1,b_1\right) \in U,a_1\in A \rbrace$,in the same way we could know $\left( B \cap U \right)$

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    Is this really what you mean? What do $A\cap U$ and $B\cap U$ mean in this context? E.g., if $A=B=\mathbb Z$, and $U=A\times B=\mathbb Z\times\mathbb Z$, then $A\cap U=\mathbb Z\cap (\mathbb Z\times \mathbb Z)=$??2012-12-27
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    @JonasMeyer: Other than naming projection maps, is there a standard say of denoting $\{ a \in A\, :\, (a,b) \in U\ \text{for some}\ b \in B\}$? I presume this is what $A \cap U$ denotes.2012-12-27
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    What does $A\cap U$ means ? $U$ is a set of ordered pairs, $A$ is not, so the intersection is always an empty set2012-12-27
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    It is standard in a direct product $A \times B$ to identify $A$ and $B$ with the subgroups $\{(a,1) \mid a \in A\}$ and $\{(1,b) \mid b \in B \}$ of $A \times B$.2012-12-27
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    @Clive: That is one possible interpretation. Notice that Derek has another interpretation, which would have been my guess if I had to guess. But if user53587 won't tell us what it means, I have no interest in guessing.2012-12-27
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    @JonasMeyer I am sorry for missing some context of them2012-12-28
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    @JonasMeyer ZFC is crappy foundation in this regard.2012-12-28

2 Answers 2

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Think about it intuitively.

Take $A,B = \mathbb{R}$. The LHS $U$ is "some set of points in the plane" whereas the RHS is "all possible x-coordinates from points in $U$ with all possible y-coordinates from points in $U$".

Clearly $U\subseteq$ RHS but the RHS can be bigger.

For a specific example let $U = \{(0,1),(1,0)\}$. Then the RHS is $\{(0,0),(0,1),(1,0),(1,1)\}$.

The point is that the object on the RHS is combining things componentwise that might not have existed when we chose "special" points to be in $U$.

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    If you like it then choose it as the best answer by clicking on the tick at the side.2012-12-28
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    But $U$ is not a subgroup of $A \times B$.2012-12-28
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    I didn't know we were looking for groups with this property. To be fair the question had been edited. Anyway, it is easy to tweak this example to provide subgroups which dont work...it is really the fact that it doesn't work as sets that makes it not work as groups.2012-12-29
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    @fretty: There is a convention widely used in the group theory community that if $G$ is a group then $H \le G$ means that $H$ is a subgroup of $G$, whereas $H \subseteq G$ means that it is a subset and not necessarily a subgroup. But of course not everyone might be aware of that!2012-12-29
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    Oh I didn't know that, I always thought it was just convention which symbol to use when writing maths on a computer. Anyway, the property is highly unlikely to hold for sets so it is easy to tweak and find groups for which it doesn't hold.2012-12-30
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How about a simple example such as $A=B\ne1$ and $U=\{(x,x)\mid x\in A\}$?

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    Is this really an example of when they aren't equal? Here $(A \cap U) = A$ and $(B\cap U) = B = A$ so the RHS is $A\times A$ which is exactly $U$.2012-12-28
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    @fretty: That is incorrect. $U\neq A\times A$. For example, if $x\neq 1$ and $1$ is the identity in $A$, then $(x,1)\in (A\times A)\setminus U$.2012-12-28
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    Stupid me, I think I might have had an off day with this question.2012-12-29