Some Notes on Relative Commutators

Let be a group and ∈ ( ). An α-commutator of elements , ∈ is defined as [ , ] = . In 2015, Barzegar et al. introduced an α-commutator of elements of and defined a new generalization of nilpotent groups by using the definition of αcommutators which is called an α-nilpotent group. They also introduced an α-commutator subgroup of , denoted by ( ) which is a subgroup generated by all α-commutators. In 2016, an α-perfect group, a group that is equal to its α-commutator subgroup, was introduced by authors of this paper and the properties of such group was investigated. They proved some results on α-perfect abelian groups and showed that a cyclic group of even order is not αperfect for any ∈ ( ). In this paper, we may continue our investigation on α-perfect groups and in addition to studying the relative perfectness of some classes of finite -groups, we provide an example of a non-abelian α-perfect 2-group.


INTRODUCTION
In 1994, an auto-commutator [ , ] = of elements G x  and ) (G Aut   was introduced by Hegarty, [1]. If is an inner automorphism such that = then auto-commutator , = is the ordinary commutator of two elements , ∈ . Hegarty generalized the definition of the center of G, One can check that L(G) is an characteristic subgroup of G which is contained in Z (G). He also introduced the autocommutator subgroup of G, denoted by K(G), which is a characteristic subgroup generated by all auto-commutators. Clearly, the commutator subgroup ′ is contained in K(G). Investigation of the relative commutators are interesting for some authors, for instance Barzegar is a normal subgroup of G that is contained in K(G). Authors of [2] also introduced a new generalization of a nilpotent group , which is called an α-nilpotent group for a fixed automorphism α of G. Here, we may present the definition of an α-nilpotent group . We start by the definition of a lower central α-series. Put is a normal subgroup of G which is invariant under α and . Following normal series is called a lower central α-series .
is called an α-nilpotent group of nilpotency class n if Clearly, if α is considered as the identity automorphism, then an α-nilpotent group is the ordinary one. In [2], it was proved that an α-nilpotent group is nilpotent, but the converse is not valid in general. For instance, authors proved that the cyclic group of order t p p p n ... Authors of [3] continued investigation on α-nilpotent groups and proved some new results on this new concept. For example, they proved that an extra special -group, is an odd prime number, is nilpotent with respect to a non-identity automorphism α but is not nilpotent relative to all its automorphisms. For an inner automorphism we can see that nilpotency and -nilpotency are equivalent. Therefore, we may ask the following question.
Question. Is there a non-inner automorphism α of nilpotent group G such that G is α-nilpotent?
This question was answered for finitely generated abelian groups, for more details see [3]. Actually, authors classified all finitely generated abelian groups which are nilpotent with respect to a non-inner automorphism. Furthermore, they proved some results on relative normal and absolute normal subgroups of some classes of finite groups. In [4], they introduced an α-perfect group G, a group which is equal to its α-commutator subgroup, for a fixed automorphism α of G. If ′ is the ordinary commutator subgroup of G, then . It follows that if G is a perfect group, then it is perfect with respect to all its automorphisms. One can check that an α-nilpotent group cannot be α-perfect, but the symmetric group of order !, is an example of a non-nilpotent group where is not α-perfect, because . The relative perfectness of abelian groups was studied by authors of [4]. In this paper, we may continue our investigation on relative perfect groups and prove some new results on some classes of finite non-abelian -groups.

RELATIVE PERFECT GROUPS
In this section, we recall the definition of an α-perfect group for a fixed automorphism α. At first, we present some results on relative perfect groups that were proved in [4]. Finally, we may add some new results on non-abelian relative perfect groups.
It might be important to find all proper absolute normal subgroups of given finite group . In [3] and [4], the structure of absolute normal subgroups of some classes of finite groups were given. For instance, we have the following results.  By Lemma 2.8, we can conclude that there is no α-perfect cyclic group of even order, for all automorphisms α of such group. If is an odd prime number, then 1 ,   r r p , is perfect for each 1 < < , but it is not -perfect.
. We can write