Left gyrotransversal of a subgroup

Definition with symbols
Let $$H$$ be a subgroup of a group $$G$$. Then, a left transversal $$B$$ (viz a system of left coset representatives) of $$H$$ in $$G$$ is termed a left gyrotransversal if it satisfies all the following conditions:


 * The identity element belongs to $$B$$ (and is thus the coset representative for $$H$$)
 * $$B = B^{-1}$$. In other words, the inverse of any left coset representative is also a left coset representative
 * $$hBh^{-1} = B$$ for any $$h \in H$$, or equivalently, $$H \subseteq N_G(B)$$