Let $X$ be an infinite set, and $k \ge 1$ be a natural number. We work without the axiom of choice.

Let $G_0$ be the full symmetric group on $X$, and let $G_1$ be the full symmetric group on ${\cal P}(X)$, the power set of $X$. Both these groups act on ${\cal P}^k(X)$ (the $k$-fold power set $\mathcal{P}(\mathcal{P}(\cdots(X)\cdots)$ of $X$) in a natural way, with $G_0$ a subgroup of $G_1$. So in ${\cal P}^k(X)$ the orbits of $G_1$ split into orbits of $G_0$.

Is the following true in ZF: for all $k\ge 1$, for all distinct elements $a\neq b$ of ${\cal P}^k(X)$ that belong to the same $G_1$-orbit, there exists $\sigma \in G_1$ such that $\sigma(a)$ and $\sigma(b)$ belong to distinct $G_0$-orbits?

A positive result would have nice consequences for the model theory of Russell-Ramsey typed set theory, but I'll say nothing about that for the moment!

`fissioning' is a synonym of`

splits'. Sorry. Thank you for your attention to this question! tf $\endgroup$11more comments