- #1
mnb96
- 715
- 5
Hello,
Let's have a group G and two subgroups A<G and B<G such that the intersection of A and B is trivial.
I consider the subgroup [itex]\left\langle A^B \right\rangle[/itex] which is called conjugate closure of A with respect to B, and it is the subgroup generated by the set: [tex]A^B=\{ b^{-1}ab \;|\; a\in A,\; b\in B\}[/tex]
It is clear that [itex]A\cap \left\langle A^B \right\rangle = A[/itex].
What about [itex]B\cap \left\langle A^B \right\rangle[/itex]?
Do B and the conjugate closure [itex]\left\langle A^B \right\rangle[/itex] have trivial intersection?
Let's have a group G and two subgroups A<G and B<G such that the intersection of A and B is trivial.
I consider the subgroup [itex]\left\langle A^B \right\rangle[/itex] which is called conjugate closure of A with respect to B, and it is the subgroup generated by the set: [tex]A^B=\{ b^{-1}ab \;|\; a\in A,\; b\in B\}[/tex]
It is clear that [itex]A\cap \left\langle A^B \right\rangle = A[/itex].
What about [itex]B\cap \left\langle A^B \right\rangle[/itex]?
Do B and the conjugate closure [itex]\left\langle A^B \right\rangle[/itex] have trivial intersection?