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Contributions to the study of subgroup lattices. (English) Zbl 1360.20002
Bucharest: Matrix Rom (ISBN 978-606-25-0229-4/pbk). viii, 216 p. (2016).
As the author tells us in his preface, this book is an improved version of his habilitation thesis at the Faculty of Mathematics of Al. I. Cuza University of Iasi, Romania. So after a short introduction into the field (Chapter 1), he does not present some parts of the theory of subgroup lattices of groups but discusses in Chapter 2 (Main results) some loosely connected properties of finite groups which he has studied in the last ten years.
We give a short list of the main subjects dealt with. For this, let \(G\) be a finite group, \(L(G)\) the subgroup lattice, \(N(G)\) the lattice of normal subgroups, \(C(G)\) the poset of cyclic subgroups of \(G\), and \(L\) be a finite lattice. Section 2.1 (Basic properties and structure of subgroup lattices) contains some general results on groups \(G\) with \(L(G)\) or \(N(G)\) pseudocomplemented, breaking points in \(C(G)\), solitary subgroups and solitary quotients in \(G\), and almost \(L\)-free groups.
In Section 2.2 (Computational and probabilistic aspects of subgroup lattices) and Section 2.3 (Other posets associated to finite groups), the author tries to compute \(|L(G)|\), \(|N(G)|\), \(|C(G)|\), the subgroup commutativity degree \(\mathrm{sd}(G)\) of \(G\), that is, the probability that \(HK=KH\) for arbitrary \(H,K\leq G\), the number of factorizations \(G=HK\) where \(H,K\leq G\), and the sum and product of all \(|H|\) where \(H\in L(G)\), \(H\in N(G)\), or \(H\in C(G)\). Of course, these numbers can only be computed for groups \(G\) for which \(L(G)\) is well-known and so the author studies them mainly for abelian, Hamiltonian, and dihedral groups, for \(p\)-groups with a cyclic maximal subgroup and for groups with only cyclic Sylow subgroups.
Section 2.4 (Generalizations of subgroup lattices) is concerned with the lattice \(\mathrm{FL}(G)\) of fuzzy subgroups of \(G\) and the final Chapter 3 (Further research) contains some 50 problems on the subjects mentioned above.
20-02 Research exposition (monographs, survey articles) pertaining to group theory
20D30 Series and lattices of subgroups
20E07 Subgroup theorems; subgroup growth
20D60 Arithmetic and combinatorial problems involving abstract finite groups