2-covering numbers of some finite solvable groups

A 2-covering for a finite group $G$ is a set of proper subgroups of $G$ such that every pair of elements of $G$ is contained in at least one subgroup in the set. The minimal number of subgroups needed to 2-cover a group $G$ is called the 2-covering number and denoted by $σ_2(G).$ In \cite{gk} it is conjectured that if $G$ is solvable and not 2-generated, then $σ_2(G)=1+q+q^2,$ where $q$ is a prime power. We disprove this conjecture.

💡 Research Summary

The paper investigates the 2‑covering number σ₂(G) of finite solvable groups, a parameter introduced by Gagola and Kirtland (2025). A 2‑covering of a group G is a collection of proper subgroups such that every unordered pair of elements of G lies together in at least one subgroup of the collection. A group admits a 2‑covering if and only if it is not generated by two elements; in that case σ₂(G) denotes the smallest possible size of a 2‑covering.

In their earlier work Gagola and Kirtland proved that for a finite p‑group with minimal generating number d(G) > 2 one has σ₂(G)=1+p+p², and they conjectured that for any finite solvable group G with d(G) ≥ 3 the 2‑covering number should always be of the form

  σ₂(G)=p^{2t}+p^{t}+1

for some prime divisor p of |G| and some positive integer t. This conjecture holds for supersolvable groups and for many examples, and it would imply that σ₂(G) is always an odd integer.

The present note disproves this conjecture by constructing two infinite families of solvable groups whose 2‑covering numbers are even and do not fit the proposed formula.

Construction of the first family (Theorem 1).
Let H be the quaternion group of order 8. For any odd prime p, H can be embedded as an absolutely irreducible subgroup of GL(2,p). Consequently there exists an irreducible H‑module V of order p²; its endomorphism ring End_H(V) is the field 𝔽_p, so the dimension r = dim_{𝔽_p} V equals 2. Define

  G = V³ ⋊ H

where V³ is the direct sum of three copies of V and H acts diagonally. By a result of Gaschütz (1959) one checks that d(G)=3, so G is not 2‑generated and therefore possesses a 2‑covering.

The maximal subgroups of G fall into two types:

1. Subgroups of the form W H_v, where W is a maximal H‑submodule of V³ and v∈V³.
2. Subgroups of the form V³ K, where K is a maximal subgroup of H.

The number of maximal subgroups of type 1 is

  γ = (p^{r+1}−1)/(p−1)·p^{r} = p² + p³ + p⁴.

A detailed analysis of all possible 2‑generated subgroups X ≤ G shows that every such X is either contained in a type 1 maximal subgroup, or X lies entirely inside V³. In the latter case X cannot be covered by any type 1 maximal subgroup, but it is contained in the unique type 2 maximal subgroup V³⟨h⟩ where h∈H has order 2. Consequently any 2‑covering must contain all γ subgroups of type 1 and at least one subgroup of type 2. Hence

  σ₂(G) = γ + 1 = 1 + p² + p³ + p⁴.

Since p is odd, this value is even, providing the first infinite family of counter‑examples.

Construction of the second family (Theorem 2).
Let q = ℓ^k be a prime power and let p be an odd prime dividing q+1. Take H to be the dihedral group of order 2p. Let E be the field with q² elements, and view its additive group V = (E,+) as an H‑module by letting a∈H of order p act as multiplication by a fixed element of order p in E^×, and letting b∈H of order 2 act via the Frobenius automorphism x ↦ x^q. One checks that V is irreducible, End_H(V) ≅ 𝔽_q, and again r = 2. Set

  G = V³ ⋊ H.

The maximal subgroups of type 1 again number

  γ = q² + q³ + q⁴.

If a 2‑generated subgroup X ≤ G contains an element of H of order p, then X is contained in a type 1 maximal subgroup. However, when X contains the involution h∈H of order 2, one can choose vectors v₁, v₂∈V³ such that X = ⟨h v₁, v₂⟩ generates V³ as an H‑module. In this situation X is not contained in any type 1 maximal subgroup; the only maximal subgroup that contains X is V³⟨h⟩. Since there are exactly p distinct involutions in H, a 2‑covering must contain all p subgroups V³⟨h⟩ together with all γ subgroups of type 1. Therefore

  σ₂(G) = γ + p = q² + q³ + q⁴ + p.

Again this number need not be of the form p^{2t}+p^{t}+1; in particular it can be even when p=2, and for odd p it deviates from the conjectured pattern.

Consequences.
Both families demonstrate that σ₂(G) can assume infinitely many even values and that the simple formula σ₂(G)=p^{2t}+p^{t}+1 does not hold for all solvable groups. The conjecture of Gagola and Kirtland is therefore false in general. The paper suggests that a more nuanced description of 2‑covering numbers is required, possibly involving the structure of maximal subgroups of the semidirect product V^{t+1}⋊H and the interaction between the module V and the acting group H. Future work may aim at classifying all possible σ₂(G) for solvable groups, identifying invariants that control the parity and magnitude of the 2‑covering number, and extending the analysis to broader classes such as supersolvable or nilpotent groups.