Integrals of groups

Abstract

An integral of a group G is a group H whose derived group (commutator subgroup) is isomorphic to G. This paper discusses integrals of groups, and in particular questions about which groups have integrals and how big or small those integrals can be. Our main results are:

Completely realisable groups

“…Clearly, any subgroup of an f -realisable group is itself f -realisable. This holds, for example, for f = D. Also, we observe that completely D-realisable groups 1 are solutions of Problem 10.19 in [1]. Throughout this paper, we assume that the above groups G and H are both finite.…”

“…We conjecture that A 4 is the smallest completely D-realisable nonabelian group. The dihedral groups D 2n with n = 3, 4, 5, 6 and the dicyclic group Dic 3 are not D-realisable because each of them has a characteristic cyclic subgroup which is not contained in center (see Proposition 3.1 of [1]). The quaternion group Q 8 is D-realisable, a group H with Q 8 ∼ = D(H) being necessarily a proper semidirect product P ⋊ A, where P is a 2-group containing a normal subgroup isomorphic to Q 8 and having D(P ) cyclic of order 2 or 4, and A is an abelian group of odd order 3 .…”

“…Since P and H/P are of coprime orders, the Schur-Zassenhaus theorem leads to H ∼ = P ⋊ A, where A ∼ = H/P is abelian of odd order. Moreover, D(P ) is a proper subgroup of D(H) because Q 8 is not p-integrable (see Theorem 4.2 of [2]) and so it is cyclic of order 2 or 4.…”

“…For each of these constructions, there is an inverse problem: Given a group G, is there a group H such that G ∼ = f (H)? (1) Several new results related to this problem have been obtained in [1,2,6,9,10] for f (H) = D(H), the derived subgroup of H. Note that in these papers the group H with the property G ∼ = D(H) has been called an integral of G by analogy with calculus. Moreover, we recall Problem 10.19 in [1] that asks to classify the groups in which all subgroups are integrable.…”

On some questions related to integrable groups

“…Recently several articles ( [1,2,12]) have appeared on the topic of the integrability of a group G, where we say that a group H is an integral of a group G if G H and we then say that G is integrable. Given two groups G ≤ U , we say that G is (relatively) integrable within U if there exists a subgroup H ≤ U such that H = G. Burnside [5] was the first to consider integrals of groups, showing, for example, that a nonabelian finite p-group with cyclic center cannot be relatively integrable within a finite p-group.…”