Similar group actions

Definition

Symbol-free definition

Two groups acting on two sets are said to act similarly if all of the following conditions are met:

• The groups are isomorphic
• There is a bijection between the sets
• For any element of the first group and the first set, applying the group action in the first group and then the isomorphism into the second group is the same as applying the isomorphism and then applying the group action in the second group.

Definition using symbols

($A$ on $\Delta$) is similar to ($B$ on $\Gamma$) if:

• There is an isomorphism $\varphi :A\rightarrow B$
• There is a bijection $\omega :\Delta \rightarrow \Gamma$
• For all $a\in A$ and $\delta \in \Delta$, we have $\varphi (a\delta )=\varphi (a)\omega (\delta )$

Examples

• $D_{3}$ on vertices on the triangle acts similarly to $S_{3}$ on the set $\{1,2,3\}$