Let $G$ and $H$ be subgroups of the symmetric groups $\mathfrak S_m$ and $\mathfrak S_n$. Assume that $n>1$ and that $H$ is a 'connected' permutation group, that is, there is no non-trivial $H$-stable subset $X\subset [n]$ (with $[n]=\{1,\dots,n\}$) such that $H$ is isomorphic to the direct product of the image of its action on $X$ and the image of its action on $[n]\setminus X$.

For example, the subgroup of $\mathfrak S_5$ generated by $(1,2)(3,4,5)$ is not connected, because it is the direct product of $\langle(1,2)\rangle\lt\mathfrak S_2$ and $\langle(3,4,5)\rangle\lt\mathfrak S_{\{3,4,5\}}$.

This terminology is from http://oeis.org/A005226 and [1].

- Question: Is there a more standard name for this property?

My original question concerns the observation that, apparently, $G\wr H = G^n \rtimes H$ is a connected subgroup of $\mathfrak S_{mn}$, provided that $H$ is connected. Note that, evidently, if $H$ is not connected, $G\wr H$ is not connected either.

Perhaps this appears more natural when using the language of combinatorial species: suppose that $\mathcal G$ is a molecular species (that is, cannot be written as a sum) and $\mathcal H$ is an atomic species (so it is molecular and additionally cannot be written as a product), and $\mathcal H\neq \mathcal X$. Then $\mathcal H\circ \mathcal G$ is atomic.

It seems to me that this has been overlooked in the literature on species. I admit that I did not try to prove it yet - I am mostly interested in a reference.

- Question: Is this known in terms of group actions?

[1] *Naughton, L.; Pfeiffer, G.*, **Integer sequences realized by the subgroup pattern of the symmetric group.**, J. Integer Seq. 16, No. 5, Article 13.5.8, 23 p. (2013). ZBL1288.20002.

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