Group Action on Subgroup by Left Regular Representation

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Theorem

Let $G$ be a group.

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


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

$\forall \left({h, g}\right) \in H \times G: h * g = \lambda_h \left({g}\right)$

where $\lambda_h \left({g}\right)$ is the left regular representation of $g$ by $h$.

Then $*$ is a group action.


Proof

The group action axioms are investigated in turn.

Let $h_1, h_2 \in H$ and $g \in G$.


Thus:

\(\displaystyle h_1 * \paren {h_2 * g}\) \(=\) \(\displaystyle h_1 * \paren {\map {\lambda_{h_2} } g}\) Definition of $*$
\(\displaystyle \) \(=\) \(\displaystyle h_1 * \paren {h_2 \circ g}\) Definition of Left Regular Representation
\(\displaystyle \) \(=\) \(\displaystyle \map {\lambda_{h_1} } {h_2 \circ g}\) Definition of $*$
\(\displaystyle \) \(=\) \(\displaystyle h_1 \circ \paren {h_2 \circ g}\) Definition of Left Regular Representation
\(\displaystyle \) \(=\) \(\displaystyle \paren {h_1 \circ h_2} \circ g\) Group Axiom $G \, 1$: Associativity
\(\displaystyle \) \(=\) \(\displaystyle \map {\lambda_{h_1 \circ h_2} } g\) Definition of Left Regular Representation
\(\displaystyle \) \(=\) \(\displaystyle \paren {h_1 \circ h_2} * g\) Definition of $*$

demonstrating that Group Action Axiom $GA \, 1$ holds.


Then:

\(\displaystyle e * g\) \(=\) \(\displaystyle \map {\lambda_e} g\) Definition of $*$
\(\displaystyle \) \(=\) \(\displaystyle e \circ g\) Definition of Left Regular Representation
\(\displaystyle \) \(=\) \(\displaystyle g\) Group Axiom $G \, 2$: Identity

demonstrating that Group Action Axiom $GA \, 2$ holds.


The group action axioms are thus seen to be fulfilled, and so $*$ is a group action.

$\blacksquare$


Also defined as

Some sources denote a mapping by placing the symbol defining that mapping on the left of its operand.

Thus under such a convention:

$\map {\lambda_h} a$ is written $a \lambda_h$

and:

$h * a$ is written $a * h$ (or even $a h$ in sources which do not place a high regard on clarity).

Thus for the left regular representation to be defined as being a group action there exists the need to use it on the inverse:

\(\displaystyle \paren {a * h_2} * h_1\) \(=\) \(\displaystyle \paren {a \lambda_{h_2^{-1} } } * h_1^{-1}\) Definition of $*$
\(\displaystyle \) \(=\) \(\displaystyle a \lambda_{h_2^{-1} } \lambda_{h_1^{-1} }\) Definition of Left Regular Representation
\(\displaystyle \) \(=\) \(\displaystyle a * \paren {h_1 \circ h_2}^{-1}\) Definition of $*$

Such a treatment can be found in 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.).

As it runs contrary to the conventions used on $\mathsf{Pr} \infty \mathsf{fWiki}$, beyond its mention here it will not be used.


However, this example indicates how the arbitrary nature of notational conventions can cause the details of results to be equally arbitrarily dependent upon the convention used.


Also see


Sources