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On the Complete Reduction of any Transitive Permutation ‐ Group; and on the Arithmetical Nature of the Coefficients in its Irreducible Components
Author(s) -
Burnside W.
Publication year - 1905
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s2-3.1.239
Subject(s) - arithmetic function , transitive relation , mathematics , permutation (music) , combinatorics , group (periodic table) , permutation group , citation , arithmetic , discrete mathematics , computer science , library science , philosophy , chemistry , organic chemistry , aesthetics
THE present paper consists of two quite distinct parts, though the second is closely connected with the first. If a group of finite order N is represented as a regular permutation-group in N symbols, and if this permutation-group is completely reduced, each irreducible component occurs a number of times equal to the number of symbols on which it operates. This result is due to Herr Frobenius. I have recently given a direct proof* of it, depending only on two fundamental properties of finite groups. The first part of the present paper determines the number of times that any give 1 irreducible component occurs, when any representation of a group o: finite order as a transitive permutation group is completely reduced. , he result, of course, involves the above mentioned theorem as a particular case. In the second part of the paper the actual reduction of the permutation-group is dealt with: first, on the supposition that the domain of rationality is defined by the characteristics; secondly, that it is defined by the roots of unity of which the characteristics are functions. Conditions are obtained under which it is possible to exhibit the completely reduced groups so that the coefficients in their transformations shall be (i) rational functions of the characteristics, (ii) rational functions of roots of unity.