Conjectures regarding the covering of groups by cosets

Let be a group, and subgroups of , and a selection of elements of . Suppose that $$G = \bigcup_{k \in \{1, \ldots, n\}} g_k H_k.$$ Then it is a theorem that for some (Lemma 4.1 here, linked in this math overflow answer).

I have some remarks/conjectures I want to record before I dig into the proof; this blog post serves that purpose, and if I make headway on understanding the proof and/or the conjectures, that may be a future blog post.

In the case when is finite, this theorem follows from a classic application of the pigeonhole principle, and in fact several much stronger claims are true. To start with, let inequality be $$\sum_{k=1}^n \frac{1}{[G : H_k]} \ge 1,$$ because , and the sizes of these cosets must add up to at least .

Conjecture 1: Inequality holds even if is infinite. When is an infinite cardinal, we take to be .

We can do better still than this; the inclusion-exclusion principle states, going back to the case of being finite, equation
$$|G| = \sum_{\ell=1}^n (-1)^{\ell+1} \left(\sum_{ 1 \le k_1 < k_2 < \cdots < k_\ell \le n} \left| g_{k_1}H_{k_1} \cap \cdots \cap g_{k_\ell} H_{k_\ell}\right| \right).$$

Lemma 2: is either empty or the coset of a subgroup of .

Proof: Consider the case and proceed by induction. Suppose the intersection is nonempty, and let . Then it’s a standard elementary fact that , so $$g_{k_1} H_{k_1} \cap g_{k_2} H_{k_2} = g H_{k_1} \cap g H_{k_2} = g(H_{k_1} \cap H_{k_2})$$ expresses the intersection as a coset.

Definition: Let’s define the abuse of notation $$\frac{1}{[G:g_{k_1}H_{k_1} \cap \cdots \cap g_{k_\ell} H_{k_\ell}]}$$ to be zero if is empty, and if . This definition makes sense for infinite, and is well-defined, because if , then is a conjugate of so has the same index.

Dividing equation by thus gives equation , $$\sum_{\ell=1}^n (-1)^{\ell+1} \left(\sum_{ 1 \le k_1 < k_2 < \cdots < k_\ell \le n} \frac{1}{[G: g_{k_1}H_{k_1} \cap \cdots \cap g_{k_\ell} H_{k_\ell}]} \right) = 1,$$ known to be valid for finite.

Conjecture 3: Equation is valid for infinite.

Some miscellaneous conjectures.

Conjecture 4: If subgroups of have finite index and have infinite index and $$G = \bigcup_{k \in \{1, \ldots, n\}} g_k H_k $$, then $$G = \bigcup_{k \in \{1, \ldots, \ell\}} g_k H_k .$$

That is, infinite index subgroups are rendered redundant in such a covering.

Conjecture 5: If and are disjoint cosets, then there exists a subgroup such that and for .

I expect Conjecture 5 to be false; when all objects considered are affine subspaces of a vector space, it is easy to see that disjoint affine subspaces can be embedded in parallel affine subspaces, and I expect the same can be done in the case when is commutative, but not when fails to be commutative.

There are some natural conjectures one can make in considering covers of a group by a class of subgroups indexed by an arbitrarily large set, but I’ll leave those conjectures unstated for now.