A Decomposition Theory for Finite Groups with Applications to P-Groups

Citation

Weichsel, Paul Morris (1960) A Decomposition Theory for Finite Groups with Applications to P-Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/FCJW-5875. https://resolver.caltech.edu/CaltechETD:etd-07072006-085918

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Let [...] be a set of finite groups and define [...] to be the intersection of all sets of groups which contain [...] and are closed under the operations of subgroup, factor group and direct product. The equivalence relation defined by [...] if [...] = [...] is studied and it is shown that if Qn and Dn are the generalized quaternion group of order 2n and the dihedral group of order 2n then [...] = [...]. A group G is called decomposable if [...] with [...] the set of proper subgroups and factor groups of G. It is shown that if G is decomposable then G must contain a proper subgroup or factor group whose class is the same as the class of G and one whose derived length is the same as the derived length of G. The set of indecomposable p-groups of class two are characterized and for [...] their defining relations are compiled. It is also shown that if the exponent of G is p and the class of G is greater than two then G is decomposable if G/Z(G) is a direct product. Finally the equivalence relation given above is modified and its connection with the isoclinism relation of P. Hall is investigated. It is shown that for a certain class of p-groups this relation is equivalent to isoclinism

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