The Left and Right Regular Group Actions of a Group on Itself

# The Left and Right Regular Group Actions of a Group on Itself

 Definition: Let $(G, \cdot)$ be a group. The Left Regular Group Action of the group $G$ on the set $G$ is the left group action of $G$ to the set$A = G$ defined for all $g \in G$ and for all $a \in A$ by $(g, a) \to g \cdot a$. The Right Regular Group Actoin of the group $G$ on the set $G$ is the right group action of $G$ to the set $A = G$ defined for all $g \in G$ and for all $a \in A$ by $(g, a) \to a \cdot g$.

If $(G, +)$ is instead an additive group it is conventional to write the left regular group action of $G$ on the set $G$ by $(g, a) \to g + a$, and the right regular group action of $G$ on the set $G$ by $(g, a) \to a + g$.

The left regular group action of the group $G$ on the set $G$ is indeed a left group action of $G$ on the set $G$ since for all $g_1, g_2 \in G$ and for all $a \in A$ we have that:

(1)
\begin{align} \quad g_1(g_2a) = g_1(g_2 \cdot a) = g_1 \cdot (g_2 \cdot a) = (g_1 \cdot g_2) \cdot a = (g_1 \cdot g_2)a \end{align}

And for all $a \in A$ we have that:

(2)
\begin{align} \quad 1a = 1 \cdot a = a \end{align}

A similar argument can be made to show that the right regular group action of the group $G$ on the set $G$ is also indeed a right group action of $G$ on the set $G$.