On Fischer subgroups of finite groups - Mao , Yu.; Ma, X.; Vorob'ev N. Т.; Karaulova, T. B. (Ukrains’kyi Matematychnyi Zhurnal, 2021)
UDC 512.542 Let F be a Fitting set of a group G, π be a nonempty set of primes, and L≤ G. In this case, F is called a Fischer π-set of G if conditions LF, KL, and H/K is a p-subgroup of L/K for a prime pπ imply necessarily that H F. It is said that a subgroup F of G is a Fischer F-subgroup of G if the following conditions hold: 1) F F; 2) if L is an F-subgroup of G normalized by F, then L≤ F. It is said that a Fitting set F of G is π-saturated if F = {H ≤ G : H/H_F E_π’ }, where E_π’ is the class of all π’-groups. In this paper, under the condition that F is a π-saturated Fischer π-set of a π-soluble group G, we prove that a subgroup V of G is an F-injector of G if and only if V is a Fischer F-subgroup of G containing a Hall π’-subgroup of G.