# Isomorphism of group actions

Let $G$ be a group acting on a set $X$, and let $H$ be a group acting on a set $Y$. I'm wondering if there is standard terminology or notation to express the following condition: there exist an isomorphism $\varphi:G\rightarrow H$ and a bijection $\sigma:X\rightarrow Y$ such that $$\sigma(gx)=\varphi(g)\sigma(x)$$ for all $g\in G$ and all $x\in X$. Basically, I want to say that not only are $G$ and $H$ isomorphic as abstract groups, but that they act in the same way on the corresponding sets.

• Such a map $\sigma$ is sometimes called a $\varphi$-equivariant isomorphism even when $\varphi$ is not an isomorphism. Commented Jun 22, 2017 at 1:49

## 2 Answers

Let $X$ and $Y$ be a left $G$-set and a left $H$-set, respectively. Then, a homomorphism of group actions is a pair $(\varphi,\sigma)$, with $\varphi:G\to H$ being a group homomorphism and $\sigma:X\to Y$ being a function compatible with the group actions in the sense that $$\sigma(g\cdot x)=\varphi(g)\cdot\sigma(x)$$ for all $g\in G$ and $x\in X$. See for example Definition 2.1 on Page 115 of Groups and Computation II by L. Finkelstein and W. M. Kantor. What you have is just an isomorphism of group actions.

If $G=H$ and $\varphi=\text{id}_G$, then $\sigma:X\to Y$ satisfying $$\sigma(g\cdot x)=g\cdot \sigma(x)$$ for all $g\in G$ and $x\in X$ is usually called a homomorphism of (left) $G$-sets. In other words, if $\sigma:X\to Y$ is a $G$-set homomorphism, then $\left(\text{id}_G,\sigma\right)$ is an example of a homomorphism of group actions from $(G,X)$ to $(G,Y)$.

• I do wonder if we have a homomorphism $(\varphi,\sigma)$ from $(G,X)$ to $(G,Y)$, where $G$ is a group homomorphism from $G$ to $G$, then is it possible to get a homomorphism $(id_G,\sigma^\prime)$ from $(G,X)$ to $(G,Y)$? Commented Oct 25, 2022 at 5:36

An action of $G$ on $X$ is given by a group homomorphism $G \to Sym(X)$.

Therefore, the action of $G$ on $X$ is isomorphic to the action of $H$ on $Y$ exactly when the diagram below commutes: $$\begin{array}{ccc} G & \to & Sym(X) \\ \downarrow & & \downarrow \\ H & \to & Sym(Y) \end{array}$$ The horizontal arrows are induced by the actions, the vertical arrow on the left is $\varphi$, and the vertical arrow on the right is induced by $\sigma$.

The diagram can be used to define morphisms of group actions, not just isomorphisms.

• What does it mean for a diagram to commute? Commented Mar 28, 2023 at 11:17
• @samerivertwice, see en.wikipedia.org/wiki/Commutative_diagram
– lhf
Commented Mar 28, 2023 at 11:21