AN EQUIVALENT DEFINITION OF A \(\mathcal{C}\)-GROUP

Abstract

In 1974, Fattahi [2] classified finite non-nilpotent groups in which every subgroup is either normal or abnormal.  Surely, it is an interesting topic to define and classify a bigger class of groups containing those classified by Fattahi.  In 2012, the first author etc defined a \(\mathcal{C}\)-group.  A finite group G is said to be a \(\mathcal{C}\)-group if for each divisor d of the order of G, G always contains a subgroup H of order d such that H is either normal or abnormal in G.  The class of \(\mathcal{C}\)-groups is really a class of groups bigger than those classified by Fattahi.  During the first author etc investigate the structure of \(\mathcal{C}\)-groups, they found that some property cannot hold if ’normal’ is changed into ’subnormal’.  So the authors of this paper is motivated to find an equivalent definition of a \(\mathcal{C}\)-group, which can be described by ’subnormal’ and ’abnormal’. A \(\mathcal{C}\)₁-group is defined , some good properties of a \(\mathcal{C}\)₁-group are given, then it is proved that a \(\mathcal{C}\)₁-group is equivalent to a \(\mathcal{C}\)-group.  At last, as an example of effectiveness of a \(\mathcal{C}\)₁-group, the necessary and sufficient conditions of a semi-product of a p-group and a p′-group to be a \(\mathcal{C}\)-group is given.