Classification Of Finite 2-groups Containing A Unique Inner Abelian Maximal Subgroup

LetNandFbe finitep-groups. If there exists a finitep-groupGhaving a normal subgroupNwithN～NandG/N～=F, thenGis calleda extension ofFbyN. IfN≤Z(G), thenGis called a central extensionofFbyN. In particular, if |N| = 2, thenGis called a central extentionofFof degree 2. In this paper, we classify finite 2-groups containing aunique inner abelian maximal subgroups by means of a central extension.