ON STRONGLY SUPERSOLUBLE FINITE GROUPS

Throughout this report, all groups are finite. The notion of a normal subgroup takes a central place in the theory of groups. One of its generalizations is the notion of a modular subgroup, i.e. a modular element (in the sense of Kurosh [1, Chapter 2, p. 43]) of a lattice of all subgroups of a group. Recall that a subgroup M of a group G is called modular in G, if the following assertions hold: 1) ⟨X,M ∩ Z⟩ = ⟨X,M⟩ ∩ Z for all X ≤ G,Z ≤ G such that X ≤ Z, and 2) ⟨M,Y ∩ Z⟩ = ⟨M,Y ⟩ ∩ Z for all Y ≤ G,Z ≤ G such that M ≤ Z. Properties of modular subgroups were studied in the book [1]. Groups with all subgroups are modular were studied by R. Schmidt [1], [2] and I. Zimmermann [3]. By parity of reasoning with subnormal subgroup, in [3] the notion of a submodular subgroup was introduced. Definition 1 [3]. A subgroup H of a group G is called submodular in G, if there exists a chain of subgroups H = H0 ≤ H1 ≤ . . . ≤ Hs−1 ≤ Hs = G such that Hi−1 is a modular subgroup in Hi for i = 1, . . . , s. Using this notion we introduce a key notion of this report. Definition 2. A group G we will call strongly supersoluble if G is supersoluble and every Sylow subgroup of G is submodular in G. Denote sU the class of all strongly supersoluble groups. The following results are obtained. Theorem 1. Let G be a group. Then the following hold: 1) if G is strongly supersoluble, then every subgroup of G is strongly supersoluble; 2) if G is strongly supersoluble and N G, then G/N is strongly supersoluble; 3) if Ni G and G/Ni is strongly supersoluble for i = 1, 2, then G/N1 ∩ N2 is strongly supersoluble; 4) if Hi G, Hi is strongly supersoluble, i = 1, 2 and H1 ∩H2 = 1, then H1 ×H2 is strongly supersoluble; 5) if G/Φ(G) is strongly supersoluble, then G is strongly supersoluble; 6) the class of groups sU is a hereditary saturated formation. We denote B the class of all abelian groups of exponent free from squares of primes. Theorem 2. The class of all strongly supersolubility groups is a local formation and has a local screen f such that f(p) = A(p− 1) ∩B for any prime p. Theorem 3. Let the group G = AB be the product of nilpotent subgroups A and B. If A and B are submodular in G, then G is strongly supersoluble. In Theorem 3 we can’t discard the submodularity of one of subgroups. Example. In group G = AB, where A ≃ Z17 and B ≃ Aut(Z17) ≃ Z16, the subgroup A is submodular, but the subgroup B is not submodular in G. The group G is supersoluble, but not strongly supersoluble. The example also shows that sU = U. Theorem 4. A group G is strongly supersoluble if and only if G is metanilpotent and any Sylow subgroup of G is submodular in G.