C-Supplemented subgroups of finite groups

Abstract

A subgroup H of a group G is said to be c-supplemented inG if there exists a subgroup K of G such thatHK=G and H \cap K is contained in Core_G (H). We follow Hall's ideas to characterize the structure of the finite groups in which every subgroup is c-supplemented. Properties of c-supplemented subgroups are also applied to determine the structure of some finite groups.

On 𝑐-supplemented maximal and minimal subgroups of Sylow subgroups of finite groups

“…By adding a saturated formation and conditions of complementation and normality, further extensions have been found: [7], [8], [9] for c-normality and [10], [11], [12] for ccomplements. In 1996 Wang generalized the above two results by replacing the normality with the weaker c-normality [7].…”

“…By minimizing the number of c-normal maximal or minimal subgroups, Wei in 2001 extended the results further to a saturated formation containing the class of supersolvable groups [8]. Wang, BallesterBolinches and Guo in 2000 introduced the concept of c-supplementation of a finite group which is weaker than both c-normality and complementation [10], [11] and generalized the above two results as applications. In [12], Wang, Wei and Li extended the results further to a saturated formation containing the class of supersolvable groups by limiting the c-supplementation of maximal or minimal subgroups to the Fitting subgroup of a solvable group.…”

Finite p–nilpotent groups with some subgroups c–supplemented

“…Also by a terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700008624 Xiuyun Guo and K. P. Shum [4] result of Schmidt [12, Theorem 9.1.9 and Exercises 9.1.11], we see that G is of order p a qP, where q is a prime distinct from/?, P is normal in G and any Sylow g-subgroup Q of G is cyclic. Furthermore, P is of exponent p if p is odd and of exponent at most 4 if p = 2.…”

THE $c$-SUPPLEMENTED PROPERTY OF FINITE GROUPS

“…Clearly, U ⊆ F, but F is not saturated. Set G = SL (2,3), H = Z(G). Then G/H is isomorphic to A 4 and so G/H ∈ F. The other hypotheses in the theorems are satisfied since H is of order 2, but G does not belong to F.…”

“…Hence, p = 2 and G F has exponent 4. At the same time, generate GL (2,3), and the following relations hold:…”