What does this group theory notation mean: $\cap_{(x \in G)}xHx^{-1}$?

Question

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?

Answer 1

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 $H$.

Answer 2

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).

Answer 3

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