arXiv:2008.13675v1 [math.GR] 31 Aug 2020 Integrals of groups II Jo˜ ao Ara´ ujo ∗ , Peter J. Cameron † , Carlo Casolo, and Francesco Matucci ‡ August 2020 In April 2018, Carlo Casolo sent the other authors detailed answers to some of the questions in the first version of the paper [1], and we immediately invited him to join us. He was very dedicated and curious about integrals and inversegroup theory problems. In fact, the current paper is in large part Carlo’s work, together with the fruits of a meeting in Florence in February 2020. Carlo passed away not long after. He was very generous and kind to all of us and is sorely missed. We dedicate this paper to his memory. Abstract An integral of a group G is a group H whose derived group (com- mutator subgroup) is isomorphic to G. This paper continues the in- vestigation on integrals of groups started in the work [1]. We study: • A sufficient condition for a bound on the order of an integral for a finite integrable group (Theorem 2.1) and a necessary condition for a group to be integrable (Theorem 3.2). • The existence of integrals that are p-groups for abelian p-groups, and of nilpotent integrals for all abelian groups (Theorem 4.1). * Universidade Aberta, R. Escola Politecnica 147, 1269-001, Lisboa, Portugal and CEMAT-Ciˆ encias, Faculdade de Ciˆ encias, Universidade de Lisboa, 1749-016, Portugal joao.araujo@uab.pt † School of Mathematics and Statistics, University of St. Andrews, UK and CEMAT-Ciˆ encias, Faculdade de Ciˆ encias, Universidade de Lisboa, 1749-016, Portugal, pjc20@st-andrews.ac.uk ‡ Dipartimento di Matematica e Applicazioni, Universit` a di Milano - Bicocca, Italy, francesco.matucci@unimib.it 1