Theory of groups of finite order

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Monograph

Persistent identifier:: 1927821487

Author:: Vitry, Urbain

Title:: Le Vignole De Poche, Ou Mémorial Des Artistes, Des Propriétaires Et Des Ouvriers

Scope:: 1 ungezähltes Blatt, 48 Seiten, 1 ungezähltes Blatt, 54 Blätter

Edition title:: Cinquième Édition, Stéréotype, Augmentée de 20 planches dessinées, cotées, et gravées sur acier Et dans laquelle lest 35 planches des quatre éditions précédentes ont été regravées à neuf sur acier, par le même

Year of publication:: 1843

Place of publication:: Paris

Containing work:: Dictionnaire Portatif D'Architecture Civile Et Des Arts Qui En Dépendent; Le Vignole De Poche, Ou Mémorial Des Artistes, Des Propriétaires Et Des Ouvriers; Dictionnaire Portatif D'Architecture Civile Et Des Arts Qui En Dépendent : Tels que la Maçonnerie, la Charpenterie, la Menuiserie, la Serrurerie, etc., etc.; ainsi que la définition des nouveaux terms dans ces artes

Identifier (digital):: 1927821487

Write comment:: Die Vorlage enthält insgesamt 2 Werke: Le Vignole De Poche, Ou Mémorial Des Artistes, Des Propriétaires Et Des Ouvriers (von Urbain Vitry; [1] Blatt, 48 Seiten, [1] Blatt, 54 Blätter) ; Dictionnaire Portatif D'Architecture Civile Et Des Arts Qui En Dépendent : Tels que la Maçonnerie, la Charpenterie, la Menuiserie, la Serrurerie, etc., etc.; ainsi que la définition des nouveaux terms dans ces artes (von Urbain Vitry, 143 Seiten); Die einseitig bedruckten Illustrationen sind Stahlstiche

Signature of the source:: Haupt 1209

Language:: French

Additional Notes:: Die Vorlage enthält insgesamt 2 Werke: Le Vignole De Poche, Ou Mémorial Des Artistes, Des Propriétaires Et Des Ouvriers (von Urbain Vitry; [1] Blatt, 48 Seiten, [1] Blatt, 54 Blätter) ; Dictionnaire Portatif D'Architecture Civile Et Des Arts Qui En Dépendent : Tels que la Maçonnerie, la Charpenterie, la Menuiserie, la Serrurerie, etc., etc.; ainsi que la définition des nouveaux terms dans ces artes (von Urbain Vitry, 143 Seiten); Die einseitig bedruckten Illustrationen sind Stahlstiche

Publisher of the original:: Audot, Éditeur Du Propriétaire Architecte. Imprimerie De Fain Et Thunot

Usage licence:: Public Domain Mark 1.0

Editor:: Audot, Louis Eustache

Engraver:: Hibon, Auguste

Printer:: Fain, Aimé-Élisabeth; Thunot, Nicolas-Marie-Eugène

Engraver:: Hibon, Auguste

Publisher of the digital copy:: Technische Informationsbibliothek (TIB)

Place of publication of the digital copy:: Hannover

Year of publication of the original:: 2025

Document type:: Monograph

Collection:: Graphic single sheets; Monographs

Chapter

Title:: DICTIONNAIRE PORTATIF D'ARCHITECTURE CIVILE.

Document type:: Monograph

Structure type:: Chapter

Section

Title:: DICTIONNAIRE D'ARCHITECTURE CIVILE.

Document type:: Monograph

Structure type:: Section

150] OF PRIMITIVE GROUPS 197 permuted by G must be capable of division into imprimitive systems, such that the number in each system is the power of a prime. 150. Let G be a &-ply transitive group of degree n (k > 2), and let G r be that sub-group of G which keeps r (< k) given symbols unchanged, so that G r is (k — r)-ply transitive in the remaining n — r symbols. Also, let H be a self-conjugate sub-group of G, and let H r be that sub-group of H which keeps the same r symbols unchanged; so that H r is the common sub-group of H and G r . Since every operation of G r transforms both H and Gy into themselves, every operation of G r must be permutable with H r \ i.e. H r is a self-conjugate sub-group of G r . Now, if r = k — 2, Gjc—2 is doubly transitive in the n — k + 2 symbols on which it operates; it is therefore primitive. Hence, unless Hk-2 consists of the identical operation only, it must be transitive in the n — k + 2 symbols. If H,t_ 2 is the identical operation, H contains no operation, except identity, which displaces less than n — k + 3 symbols. Suppose, first, that H contains operations, other than iden tity, which leave one or more symbols unchanged. Then, since H is a self-conjugate sub-group and G is /r-ply transitive, it may be shewn, exactly as in § 138, that H must contain operations displacing not more than 2k — 2 symbols. Hence H k _ 2 can consist of the identical operation alone, only if n — k + 3 ^2k — 2, or k <j: > + f. When this inequality holds, we have seen (§ 138) that G contains the alternating group. Hence in this case, if G does not contain the alternating group, it follows that H k _ 2 is transi tive in the n — k + 2 symbols on which it operates. Since H is self-conjugate and G is &-ply transitive, H must contain a sub-group conjugate to H k ^ which keeps any other k — 2 symbols unchanged. Hence //¿_ 3 must be doubly transi tive in the n — k + 3 symbols on which it operates; and so on. Finally, if G is not the symmetric group (the alternating group, being simple, contains no self-conjugate sub-group) H must be (k — l)-ply transitive.

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