Exercise 4.3.1 (Corresponding left and right actions)

Question

Suppose $G$ has a left action on a set $A$, denoted by $g\cdot a$ for all $g \in G$ and $a\in A$. Denote the corresponding right action on $A$ by $a\cdot g$. Prove that the (equivalence) relations $\sim$ and $\sim'$ defined by
$$a\sim b \qquad 	ext{if and only if}\qquad a = g\cdot b \quad 	ext{for some } g \in G$$
and
$$a\sim' b \qquad 	ext{if and only if}\qquad a = b\cdot g \quad 	ext{for some } g \in G$$
are the same relation (i.e., $a\sim b$ if and only if $a \sim' b$).

Richard Ganaye · Accepted Answer

\begin{proof} 
Since the left and right actions are corresponding, by definition
\begin{align}
g\cdot a = a\cdot g^{-1} \qquad (a \in A,\ g \in G).
\end{align}
Let $a,b \in A$.  If $a \sim b$, then $a = g\cdot b$ for some $g \in G$, thus $a = b \cdot g^{-1}$ by (1).

Put $h = g^{-1}$. Then $h \in G$ and $a= b \cdot h$, thus $a \sim' b$. 

Similarly, if  $a \sim' b$, then $a = b\cdot g$ for some $g \in G$. If $h = g^{-1}$, then $h \in G$ and $a = b \cdot h^{-1}$, thus $h \cdot b = b \cdot h^{-1}= a$ by (1), thus $a\sim b$.

In conclusion, for all $a,b \in A$,
$$a\sim b	ext{ if and only if }a \sim' b.$$
\end{proof}

Exercise 4.3.1 (Corresponding left and right actions)

Answers

Comments

Exercise 4.3.1 (Corresponding left and right actions)

Answers

Comments

Add answer