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I'm trying to understand this proof https://math.stackexchange.com/a/128549/205193 which does not seem complicated but I don't understand why :

why $p(x)=p(y)$, implies that $x$ and $y$ are in the same left coset of $H$ and the same left coset of $K$.

for exemple, if $h(x)=2$, $k(x)=3$ and $h(y)=3, k(y)=2$ dont we have $p(x)=p(y)$ even if they arent in the same cosets?

and why do we have that:

If $x$ and $y$ are in different left cosets of $H∩K$, then $p(x)≠p(y)$

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The notation $\langle h(x),k(x)\rangle$ in that answer denotes an ordered pair (more often written $(h(x),k(x))$), so $\langle h(x),k(x)\rangle=\langle h(y),k(y)\rangle$ iff $h(x)=h(y)$ and $k(x)=k(y)$.

If $p(x)=p(y)$, then $h(x)=h(y)$, so $x^{-1}y\in H$. Similarly, $k(x)=k(y)$, so $x^{-1}y\in K$. Thus $x^{-1}y\in H\cap K$, so they are in the same left coset of $H\cap K$. This is the contrapositive of your second question.

Eric Wofsey
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  • Thanks I thought the notation was scalar product.. – guest Oct 04 '15 at 20:16
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    Ah, I see (from your example I thought you thought it was an unordered pair). Angle brackets do often mean scalar product, but they are also sometimes used for ordered pairs (or ordered $n$-tuples), and in this context the latter makes a lot more sense! – Eric Wofsey Oct 04 '15 at 20:18