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abelian group abelian subgroup abstract group alternating group automorphism bitangents characteristic subgroup coefficients commutator subgroup corresponding cyclic group cyclic subgroups determine dicyclic group dihedral group direct product distinct divisor domain equation follows G contains given group G group of degree group of isomorphisms group of order group whose order Hence identity imprimitive integer intransitive invariant subgroup involves irreducible letters linear group linear transformation matrices multiplied non-abelian group obtained operators of G operators of order order g order pm orders divide pairs permutes possible prime number primitive group proof quartic quotient group rational function replaced represented roots of unity set of conjugates similarity-transformation simple group simply isomorphic solvable by radicals solvable group stitution subgroup of G subgroup of index subgroup of order substitutions of G Sylow subgroups symmetric group theorem theory tion transitive group transitive substitution group triangle unaltered variables zero
Page 110 - ... contained in H. It results also directly from the preceding developments that if H does not include all the operators of order p in the given set of independent generators of G, then these remaining operators of order p may be combined into subsets such that all the operators of each subset generate either an abelian group of order pm and of type (1, 1, 1, . .), or a group having the properties which we proved to apply to H. Hence the theorem. // a group G contains a set of independent generators...
Page 35 - G on the same n letters constitute a group which is similar to G. To Jordan is due the fundamental concept of class of a substitution group and he proved the constancy of the factors of composition. He also proved that there is a finite number of primitive groups whose class is a given number greater than 3, and...
Page 83 - G is composed of all those in which the subgroups composed of all the substitutions of G which omit a given letter correspond to subgroups of degree n.
Page 116 - G is the direct product of the quaternion group and an abelian group of order 2° and of type (1, 1, 1, . . . ). Hence it will be assumed in what follows that G involves non-commutative operators of order 2. Every operator of order 4 contained in G is transformed either into itself or into its inverse by every operator of G and an operator of order 2 contained in G has at most two conjugates under the group.2 Let...
Page 162 - The necessary and sufficient condition that there are substitutions besides identity which are commutative with every substitution of a transitive group of degree n is...
Page 162 - It results directly from the preceding developments that a necessary and sufficient condition that the group of isomorphisms of an abelian group...
Page 127 - See Rend. Ace. Lincei, ser. 4, 1, 281 (1885). Among the results of the present paper which are supposed to be new are the following: The number of the different operators in each of the possible sets of independent generators of a group whose order is a power of a prime number is the same, — that is, if the order of a group is a power of a prime number, the number of its independent generators is an invariant of the group. The ^-subgroup of every direct product is the direct product of the ^-subgroups...
Page 90 - X independent generators is commonly denned so that the group generated by every X - 1 of these generators has only the identity in common with the group generated by the remaining operator. For an abelian group whose order is a power of a prime number the number of the different operators in a possible set of independent generators is the same under both of the given definitions of a set of independent generators. The fact that the number of independent generators of a group...
Page 238 - = XV, X being a transformation of (D). For, V'V~l must leave fixed each triangle, and is therefore a transformation X as defined. We are now in a position to construct the required groups. By direct application we verify that the transformations U} V...