Primitive Groups (Paperback)

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: ...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 its n variables is invariant under it. /. Only similarity substitutions are commutative with every substitution of an irreducible group. Let S be a substitution commutative with every substitution of a group G. There exists a substitution T of non-zero determinant that transforms S into the canonical form of 22. By 23, I and II, either TGT is reducible or T-lST is a similarity substitution. But if G is irreducible, TGT is likewise irreducible. 25. The Sum of Substitutions. It is possible to give a definition of the sum of two substitutions The sum A--B is the substitution A + B:, == (ai, + &ij), (--1. 2, n). 1=i The addition of substitutions is commutative: If C is a third substitution a C:, = 2 Cijxj (-==1, 2, -), J=i the product 04-T-B)C = /4C + BC; for (+B)Cis n n xs = 2 2 (an 4-fciJ)Cjkk 1=i k i n n n n = 22 Oiicikxl + 22 biicikxi (- = 1, 2, n). 1=i =i 1=i k=l Similarly C(A + B)=CA + CB. I. In the sum of the substitutions of an irreducible group every coefficient is zero. The sum of all the substitutions of an irreducible group is commutative with every substitution of the group and is, b...