On the Kegel–Wielandt $$\\sigma $$-problem for binary partitions

Abstract

Let $$\\sigma = \\{ {\\sigma }_{i} : i \\in I \\}$$\n be a partition of the set $${\\mathbb {P}}$$\n of all prime numbers. A subgroup X of a finite group G is called $$\\sigma $$\n -subnormal in G if there is a chain of subgroups $$\\begin{aligned} X=X_{0} \\subseteq X_{1} \\subseteq \\cdots \\subseteq X_{n}=G \\end{aligned}$$\n where for every $$j=1, \\dots , n$$\n the subgroup $$X_{j-1}$$\n is normal in $$X_{j}$$\n or $$X_{j}/Core_{X_{j}}(X_{j-1})$$\n is a $${\\sigma }_{i}$$\n -group for some $$i \\in I$$\n . In the special case that $$\\sigma $$\n is the partition of $${\\mathbb {P}}$$\n into sets containing exactly one prime each, the $$\\sigma $$\n -subnormality reduces to the familiar case of subnormality. A finite group G is $$\\sigma $$\n -complete if G possesses at least one Hall $$\\sigma _i$$\n -subgroup for every $$i \\in I$$\n , and a subgroup H of G is said to be $$\\sigma _i$$\n -subnormal in G if $$H \\cap S$$\n is a Hall $$\\sigma _i$$\n -subgroup of H for any Hall $$\\sigma _i$$\n -subgroup S of G. Skiba proposes in the Kourovka Notebook the following problem (Question 19.86), that is called the Kegel–Wielandt $$\\sigma $$\n -problem: Is it true that a subgroup H of a $$\\sigma $$\n -complete group G is $$\\sigma $$\n -subnormal in G if H is $$\\sigma _i$$\n -subnormal in G for all $$i \\in I$$\n ? The main goal of this paper is to solve the Kegel–Wielandt $$\\sigma $$\n -problem for binary partitions.