It is shown that two formations of finite groups, one was introduced by V.S. Monakhov and V.N. Kniahina and another one was introduced by R. Brandl, are coincides.In this note only finite groups are considered. V.S. Monakhov and V.N. Kniahina [1] studied the class X of all groups whose cyclic primary subgroups are P-subnormal. It was shown that X is a hereditary saturated formation and U ⊂ X ⊂ D where U and D are classes of all supersoluble groups and groups with Sylow tower of supersoluble type respectively. In [2, 3] R. Brandl studied groups satisfying some lawü k (x, y) = 1 whereü 1
In this paper, we study finite groups with some complemented subgroups of prime orders. We prove the p-supersolvability of a group with complemented subgroups of order p, where p is the second smallest prime divisor of the group order.
Notation and preliminary resultsWe consider only finite groups. We use standard terminology and notation in finite group theory which correspond to [8,9].Let be a set of prime numbers. Denote by 0 the complement to in the set of all prime numbers. The symbol is also used to denote the function defined on the set of positive integers as follows: .a/ is the set of primes dividing a positive integer a. For a group G, define .G/ D .jGj/. Cyclic and dihedral groups of Brought to you by | Harvard University Authenticated Download Date | 7/24/15 12:20 PM
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