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I came across on this website 1 the notation $\cap_{(x \in G)}xHx^{-1}$, where $H$ is a subgroup of $G$. What does this mean?

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    $\begingroup$ The answer is in a footnote at the bottom of the page you linked to. $\endgroup$
    – Barry Cipra
    May 22, 2020 at 22:30

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As a set, $$xHx^{-1}=\{xhx^{-1}\mid h\in H \}.$$ It is a nice exercise to show that $$\bigcap_{x\in G}xHx^{-1}$$ is a normal subgroup of $G$ (by construction essentially) and that it is the largest normal subgroup of $G$ contained in $H$. This thing is called the core of $H$.

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    $\begingroup$ It may worth adding that it is the kernel of the action of $G$ by left multiplication on the set of the left cosets of $H$. $\endgroup$
    – user750041
    May 23, 2020 at 6:04
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This is a big intersection of all sets $xHx^{-1}$ for all $x\in G$.

Each of those sets is actually of the form: $xHx^{-1}=\{xhx^{-1}: h\in H\}$ and is another subgroup of $G$ (a "conjugate" of $H$).

You may know that intersection of arbitrarily many subgroups of $G$ is again a subgroup of $G$, so the whole thing $\bigcap_{x\in G}xHx^{-1}$ is another subgroup of $G$. It can be shown that it is a normal subgroup of $G$ (even if $H$ isn't necessarily).

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We call this the core(H) in G. That is the largest normal subgroup of “G” contained in the subgroup H.

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