Let [Formula: see text] be a finite group and [Formula: see text] a subgroup of [Formula: see text]. Then [Formula: see text] is said to be [Formula: see text]-permutably embedded in [Formula: see text] if a Sylow [Formula: see text]-subgroup of [Formula: see text] is also a Sylow [Formula: see text]-subgroup of some [Formula: see text]-permutable subgroup of [Formula: see text] for every prime dividing the order of [Formula: see text]. We say that [Formula: see text] is nearly[Formula: see text]-embedded in [Formula: see text] if [Formula: see text] has a normal subgroup [Formula: see text] such that [Formula: see text] is [Formula: see text]-permutable in [Formula: see text] and [Formula: see text], where [Formula: see text] is the subgroup of [Formula: see text] generated by all those subgroups of [Formula: see text] which are [Formula: see text]-permutably embedded in [Formula: see text]. In this paper, we study the properties of the nearly [Formula: see text]-embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized.
A finite -group is said to have the property , if, for any abelian subgroup of , there is | ( )/ ( )| ≤ . We show that if satisfies , then has the following two types: (1) is isoclinic to some stem groups of order 5 , which form an isoclinic family. (2) is isoclinic to a special -group of exponent . Elementary structures of groups with are determined.
PreliminariesLet be a prime and be a finite -group. A group is said to have the property , if any abelian subgroup of satisfies | ( )/ ( )| ≤ . It is well known that if ( )/ ( ) is cyclic, then is abelian. It rises to consider the groups with property .Mann [1] obtained the structure of 2-groups with property ; leting any abelian subgroup of 2-group satisfy | ( )/ ( )| ≤ , then is isoclinic to a dihedral group. And for any odd prime he showed that if any abelian subgroup of satisfies | ( )/ ( )| ≤ , then / ( ) is an elementary abelian -group or a nonabelian group of order 3 and exponent .Lemma 1 (see [1] This paper further discusses groups with property . In the following all considered groups are finite -groups among which is an odd prime.First we state some notions and lemmas.Definition 2. Groups and are said to be isoclinic, if there exist isomorphisms : / ( ) → / ( ) and : → which are compatible, in the sense that ( ( )) = ( ) and ( ( ))
Main ResultsWe need the following result.
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