Distinguished set of coset representatives

Definition

Let $G$ be a group and $H$ be a subgroup of $G$. A left transversal $T$ for $H$ in $G$ (i.e., a choice of one element from each left coset of $H$ in $G$) is termed a distinguished set of coset representatives if $hth^{-1}\in T$ for any $h\in H,t\in T$.

An analogous definition holds for right coset representatives.

A subgroup of a group need not possess a distinguished set of coset representatives. In fact, a subgroup possesses a distinguished set of coset representatives if and only if it is a subset-conjugacy-closed subgroup. Further information: Equivalence of definitions of subset-conjugacy-closed subgroup

References

Journal references

• Paper:Zappa-distinguished^{More info}
• Paper:SuzukiHallnormal^{More info}