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Enumeration by frame group and range groups
Author(s) -
Redfield J. Howard
Publication year - 1984
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.3190080204
Subject(s) - mathematics , enumeration , range (aeronautics) , group (periodic table) , frame (networking) , group theory , conjugate , combinatorics , extension (predicate logic) , symmetric group , algebra over a field , pure mathematics , discrete mathematics , computer science , mathematical analysis , telecommunications , chemistry , materials science , organic chemistry , composite material , programming language
Wider extension is here given to the theory developed in an earlier paper, in which the enumeration of certain configurations or combinatorial entities was made to depend on two or more groups ( range groups ). The present paper recognizes for the first time the essential part played by an additional group, the frame group , which includes all the range groups as subgroups, and which was implicit in the earlier theory as the symmetric group of the degree of the range groups. The use of frame groups which are not symmetric greatly extends the range of application of the theory, and allows its development in terms of abstract groups. By the use of the conjugate sets of the subgroups (both cyclic and noncyclic) of the frame group, instead of merely the conjugate sets of its operations , a complete solution is obtained for the problem (solved only for special cases in the earlier paper) of determining the invariance groups admitted by the various configurations enumerated.