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A right transversal is also known as a set of right coset representatives

http://www.maplesoft.com/support/help/Maple/view.aspx?path=group/cosets

if impossible, does it mean that is condition to search all groups in order to find a permutation group which right coset has only one element?

bolo
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    All cosets have the same number of elements. If a coset has just one element it is a coset of the trivial subgroup. On the other hand if there is just one coset it is the whole group. – Mark Bennet Jul 03 '13 at 13:57

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An important and fundamental fact from algebra and the study of groups is that every coset has the same number of elements (which leads directly to the statement of Lagrange's Theorem. Lagrange's Theorem says that if $H$ is a subgroup of $G$, then the number of cosets is $\dfrac{|G|}{|H|}$, and each coset is associated to $|H|$ different elements of $G$.

So if you want to have cosets that have only one element, then you are quotienting by only the trivial subgroup. If you demand that all subgroups yield quotients that have cosets with only one element, then you must be starting with the trivial group.

davidlowryduda
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