Items related to Primitive groups Volume 1
Synopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1921 Excerpt: ...is evident from this example: The four substitutions x = x', x--ix', x----x', x----ix', (i= V--1) are arranged in one-to-one correspondence to 1, (abed), (ac)(bd), (adeb), and therefore constitute a group. We shall use the term linear group as a synonym for "group of linear substitutions" or "group of substitutions," and where no undesirable ambiguity is introduced, the word group will be used for both linear groups and groups of permutations, that is, for all linear groups. The definitions of "order of a group," "Abelian group," "cyclic group," "diedral group," "generators," "subgroup," "Sylow subgroup," "conjugates," "complete set of conjugates," "cross-cut," "central," "permutable groups," "direct product of two groups," "isomorphism," "homomorphism," "equivalent groups," "similar groups," "complete groups," may now be extended to groups of substitutions, if in each definition we replace "permutation" by "substitution." Every group is isomorphic to a regular group, and in consequence the greater part of the theorems proved in Chapters I and II hold true for linear groups. The number of variables transformed by a substitution is its degree. The number of variables transformed by the substitutions of a group may be called the degree of the group. A group of degree n is reducible when s ( n) linear functions of the variables exist which are transformed into linear functions of themselves by every substitution of the group. If this condition is not satisfied the group is irreducible. Every group of permutations is reducible because the sum of...
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