Group Action on Subgroup by Right Regular Representation is not Transitive

Theorem

Let $G$ be a group.

Let $H$ be a proper subgroup of $G$.

Let $*: H \times G \to G$ be the group action defined as:

$\forall \tuple {h, g} \in H \times G: h * g = \map {\rho_{h^{-1} } } g$

where $\map {\rho_{h^{-1} } } g$ is the right regular representation of $g$ by $h^{-1}$.

Then $*$ is not transitive.

Proof

From Group Action on Subgroup by Right Regular Representation it is established that $*$ is a group action.

From Orbit of Group Action on Subgroup by Right Regular Representation is Right Coset:

$\forall x \in G: \Orb x = H x$

where $H x$ is the right coset of $H$ by $x$.

From Right Coset Space forms Partition it is apparent that $H x \ne G$ unless $H = G$.

Thus $\Orb x \ne G$ and the result follows by definition of transitive group action.

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions