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

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?

• The answer is in a footnote at the bottom of the page you linked to. May 22, 2020 at 22:30

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 $$H$$.

• 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 $H$.
– user750041
May 23, 2020 at 6:04

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).

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