Abstract:Let $\Psi (\Sigma )$ be the Isbell-Mr\'owka space associated to the $MAD$-family $\Sigma $. We show that if $G$ is a countable subgroup of the group ${\bold S}(\omega )$ of all permutations of $\omega $, then there is a $MAD$-family $\Sigma $ such that every $f \in G$ can be extended to an autohomeomorphism of $\Psi (\Sigma )$. For a $MAD$-family $\Sigma $, we set $Inv(\Sigma ) = \{ f \in {\bold S}(\omega ) : f[A] \in \Sigma $ for all $A \in \Sigma \}$. It is shown that for every $f \in {\bold S}(\omega )$ there is a $MAD$-family $\Sigma $ such that $f \in Inv(\Sigma )$. As a consequence of this result we have that there is a $MAD$-family $\Sigma $ such that $n+A \in \Sigma $ whenever $A \in \Sigma $ and $n < \omega $, where $n+A = \{ n+a : a \in A \}$ for $n < \omega $. We also notice that there is no $MAD$-family $\Sigma $ such that $n \cdot A \in \Sigma $ whenever $A \in \Sigma $ and $1 \leq n < \omega $, where $n \cdot A = \{ n \cdot a : a \in A \}$ for $1 \leq n < \omega $. Several open questions are listed.
Keywords: $MAD$-family, Isbell-Mr\'owka space
AMS Subject Classification: 54A20, 54A35