Originally Posted by
joestevens Definition: $\displaystyle G$ acts on $\displaystyle X$ if there exists a function $\displaystyle \alpha : G \times X \rightarrow X$ such that
i) for $\displaystyle g,h \in G,\ \alpha_g \circ \alpha_h = \alpha_{gh}$
ii) $\displaystyle \alpha_e = e_X$.
Should I use i & ii above as the definition of the group action $\displaystyle G$ on $\displaystyle X$ and show $\displaystyle A$ also acts on $\displaystyle X$ by $\displaystyle a.x = \phi(a).x$?
Am I to take $\displaystyle A$ as a subgroup of $\displaystyle G$ or is $\displaystyle A$ just any group?