Theory and Applications of Finite Groups, Volume 10 |
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Common terms and phrases
a₁ abelian group abelian subgroup abstract group alternating group bitangents characteristic subgroup co-sets coefficients commutator subgroup corresponding cubic cyclic group cyclic subgroups determine dicyclic group dihedral group direct product distinct divisor domain equation factors follows G contains G₁ given group G group of degree group of isomorphisms group of order group whose order H₁ 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 24 order pm pairs permutes prime number primitive group quotient group rational function replaced represented roots of unity s₁ set of conjugates similarity-transformation simple group simply isomorphic solvable by radicals solvable group subgroup of G subgroup of index subgroup of order substitutions of G Sylow subgroups symmetric group t₁ theorem theory tion transitive group transitive substitution group triangle unaltered variables y₁ zero
Popular passages
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...
Page 82 - ... degree n and index n, then the group of isomorphisms of G can be represented as a transitive substitution group of degree n which contains G as an invariant subgroup.