By Allen Nussbaum
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Additional resources for Applied Group Theory for Chemists, Physicists and Engineers
In this case, we specify k' by k'==k modulo 5, k' is of order dividing 4 modulo powers of 5 dividing N, k' == + 1 modulo any divisor of n prime to 5. These conventions are the simplest ones that ensure that the algebraic conjugation *k' has the correct effect on nth roots of unity, while fixing 'irrelevant' irrationalities. Any rule that extends this to all numbers n must be to some extent arbitrary. The rule for proxy characters is precisely similar. The characters in the cohort C' k for which a given character X from the cohort C serves as proxy are the algebraically conjugate characters X' k ', where k' is determined just as above from n, the order of the cyclic quotient of the multiplier realized faithfully by X, and any number N for which the values of X lie in the field of Nth roots of unity.
A similar, but much smaller, error was discovered in the table for the outer automorphism group of J3 - 5. Acknowledgements This is perhaps the place for a comment about references and the bibliography. Our entries for the individual groups contain no explicit references to sources. This is because the few lines we allot to any given construction often contain material culled from many places (sometimes barely remembered) which might have been transformed beyond recognition by our own investigations or transliteratiQns.
20. Abbreviated character tables There are some groups whose character tables seemed to deserve compression before we felt able to justify their inclusion. In these cases, indicated by the phase 'Abbreviated character table', we have printed just one representative from each algebraically conjugate family of classes or characters. An entry of form NID in the centralizer order row indicates that the corresponding class is the only printed member of an algebraically conjugate family of D classes, each of centralizer order N.