# Connected permutation groups and wreath product

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].

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.

1. 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.

• I am not completely sure what you mean by a direct product of two actions. Certainly, if groups $G$ and $H$ act on sets $X$ and $Y$, then there is an induced action of $G \times H$ and $X \times Y$. But in your situation does that mean that if $H$ was the direct product of actions of $H_1$ and $H_2$, then $H = H_1 \times H_2$? (We can assume that the actions of $G$ and $H$ are faithful.) Commented Apr 25, 2019 at 10:27
• @DerekHolt: Yes, that is what I meant to write. The wording is from oeis.org/A005226. Commented Apr 25, 2019 at 11:08
• I think it must be very unusual for a wreath product to decompose nontrivially as a direct product, but of course that would need proof. Commented Apr 25, 2019 at 12:17
• @DerekHolt: well, it evidently decomposes whenever $H$ is a direct product. I am quite sure it does not decompose otherwise (provided $H$ is nontrivial). Commented Apr 25, 2019 at 13:19
• I don't believe it does decompose even when $H$ does. Let $G = \langle (1,2) \rangle$, and $H = \langle (1,2)(3,4),(1,3)(2,4) \rangle$. Then $G \wr H$ is a group of order $2^6$ that does not deompose as a direct product. Commented Apr 25, 2019 at 14:08

I am afraid that I would have no idea where to look for a reference for this statement, but here is a very rough sketch proof. I can fill in details if necessary.

Let $$A_1,\ldots,A_s$$ be the orbits of $$G$$ on $$[m]$$ and $$B_1,\ldots,B_t$$ the orbits of $$H$$ on $$[n]$$. We can assume that each $$|B_j| > 1$$, since otherwise the action of $$H$$ would be disconnected.

Then the orbits of $$W := G \wr H$$ on $$[mn]$$ can be labelled $$C_{ij}$$, $$1 \le i \le s$$, $$1 \le j \le t$$, where each $$C_{ij}$$ is a union of $$|B_j|$$ orbits of size $$|A_i|$$ of the base group $$G^n$$ of $$W$$, and the components of $$G^n$$ are acting as in the action of $$G$$ on $$A_i$$ on these orbits, and the induced action of $$H$$ on the set of orbits is the same as its action on $$B_j$$.

Suppose that the action of $$W$$ is disconnected. Then $$[mn] = X_1 \cup X_2$$ with $$W \cong W_1 \times W_2$$, and the action induced from product action sof $$W_i$$ on $$X_i$$.

Then $$X_1$$ and $$X_2$$ are unions of some of the orbits $$C_{ij}$$. Suppose that $$C_{11} \in X_1$$. The key to the proof is that $$C_{i1}$$ must lie in $$X_1$$ for all $$i$$. To see this, note that if say $$C_{2j} \in X_2$$, then the pointwise stabilizer of $$X_1$$ and hence of $$C_{11}$$ must act transitively on $$C_{21}$$. But we are assuming that $$|B_1|>1$$, and the action of $$H$$ on the base group orbits in $$C_{11}$$ is the same as its action on those of $$C_{21}$$ (i.e. action of $$H$$ on $$B_1$$) and so the pointwise stabilizer of $$C_{11}$$ must stabilize each orbit of the base group within $$C_{21}$$ and hence it cannot act transitively $$C_{21}$$.

So there is a partition of $$[n]$$ into two sets $$Y_1$$ and $$Y_2$$ such that, for $$k=1,2$$, $$X_k$$ is the union of all orbits $$C_{ij}$$ with $$j \in Y_k$$. Now it is clear by looking at the induced action of $$H$$ on the orbits of the base group that the decomposition $$W = W_1 \times W_2$$ induces a decomposition $$H = H_1 \times H_2$$ with $$H_i$$ acting on $$Y_i$$.

• Follow up question: is "connected permutation group" standard terminology, or is this a concept which doesn't arise too frequently? Commented Apr 26, 2019 at 21:41