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Questions tagged [equivariant-maps]

Questions about or involving equivariant maps, the natural maps between $G$-sets.

Let $G$ be a group, and let $X$ and $Y$ be sets with $G$-actions. A map $f : X \to Y$ is said to be equivariant if $f(g\cdot x) = g\cdot f(x)$ for all $g \in G$ and $x \in X$. That is, the following diagram commutes

$$\require{AMScd} \begin{CD} X @>{g\cdot}>> X\\ @V{f}VV @VV{f}V \\ Y @>{g\cdot}>> Y. \end{CD}$$

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On equivariant maps

It is well know the notion of Equivariant map . Now, consider the situation: $X$ and $Y$ being $G$- and $H$-sets, resp., and a homomorphism $\varphi: G\to H$. How is called a map $f:X\to Y$ satisfying $f(gx)=\varphi(g)f(x)$ ?
Sigur
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For which values of a, b, c, d do $\theta_{a,b}$ and $\theta_{c,d}$ commute?

In Sets and Groups by Green a question 5 from the chapter 3 reads: Write $\theta_{a,b}$ for the map of the preceding exercise [which is $\theta(x)=ax+b=\theta_{a,b}$]. Prove $\theta_{a,b}\theta_{c,d}=\theta_{ac,ad+b}$, for any real numbers a, b, c,…
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