Endliche Gruppen mit vielen modularen Untergruppen

On one generalization of modular subgroups

“…Therefore, every (n−1)-maximal subgroup of E is either modular or S-quasinormal in E by Lemmas 2.2(4) and 2.3 (1). Thus, by the isomorphism G/N ≃ E, Lemma 2.2(5) implies that every (n − 1)-maximal subgroup of G/N is either modular or S-quasinormal in G/N , and also we have n − 1 ≤ |π(G/N )| + r. The lemma is proved.…”

“…Take a prime divisor q of the order of G distinct from p. Take a Hall q ′ -subgroup E of G, and let E ≤ W where W is a maximal subgroup of G. Then N ≤ E and since G is soluble, Lemmas 2.2(4) and 2.3 (1) imply that the hypothesis holds for W . Consequently, the choice of G implies that for some prime t dividing |E| a Sylow t-subgroup Q of E is normal in E. Proof.…”

“…We use N n and U s to denote the classes of all nearly nilpotent and of all strongly supersoluble groups, respectively. Nearly nilpotent and strongly supersoluble groups were studied respectively in [1] and [2,3].…”

“…These two results were developed by many authors. In particular, Schmidt proved [1] that: if all 2-maximal subgroups of G are modular in G, then G is nearly nilpotent; if all 3-maximal subgroups of G are modular in G and G is not supersoluble, then either G is a group of order pq 2 for primes p and q or G = Q ⋊ P , where Q = C G (Q) is a quaternion group of order 8 and |P | = 3. Mann proved [15] that if all n-maximal subgroups of a soluble group G are subnormal and…”

Lie algebras whose maximal subalgebras are modular

“…We propose to study here the structure of them over any field of characteristic zero. The corresponding problem for groups was considered by Schmidt [6] and Venzke [11] THEOREM (Schmidt). A finite group G is an M(l)-group if and only if G is supersolvable and for each complemented chief factor H/K of G, |Aut G (H/K)| is prime (or 1).…”