﻿of 
  a 
  Substitution- 
  Group 
  to 
  itself. 
  235 
  

  

  the 
  same 
  element. 
  The 
  second 
  elements 
  in 
  each 
  of 
  the 
  cor- 
  

   responding 
  substitutions 
  may 
  be 
  made 
  identical 
  by 
  trans- 
  

   forming 
  one 
  of 
  the 
  two 
  constituent 
  groups 
  by 
  means 
  of 
  a 
  

   substitution 
  which 
  does 
  not 
  contain 
  the 
  first 
  element 
  of 
  all 
  the 
  

   substitutions. 
  As 
  the 
  groups 
  remain 
  simply 
  isomorphic, 
  and 
  

   their 
  corresponding 
  substitutions 
  coincide 
  with 
  respect 
  to 
  

   their 
  first 
  two 
  elements, 
  they 
  must 
  coincide 
  throughout. 
  

   Hence, 
  every 
  simple 
  isomorphism 
  of 
  R 
  to 
  itself 
  may 
  be 
  

   obtained 
  by 
  writing 
  after 
  each 
  substitution 
  of'H 
  its 
  transform 
  

   with 
  respect 
  to 
  a 
  given 
  substitution 
  which 
  does 
  not 
  contain 
  an 
  

   arbitrary 
  element 
  of 
  R. 
  

  

  All 
  the 
  substitutions 
  which 
  transform 
  R 
  into 
  itself, 
  and 
  

   contain 
  only 
  elements 
  of 
  R, 
  form 
  a 
  transitive- 
  group 
  of 
  

   degree 
  n, 
  n 
  being 
  the 
  order 
  of 
  R. 
  A 
  subgroup 
  of 
  this 
  

   group, 
  which 
  contains 
  all 
  its 
  substitutions 
  which 
  do 
  not 
  

   contain 
  a 
  given 
  one 
  of 
  its 
  n 
  elements, 
  cannot 
  contain 
  any 
  

   substitution 
  besides 
  identity 
  that 
  is 
  commutative 
  to 
  every 
  

   substitution 
  of 
  R. 
  Hence, 
  no 
  two 
  substitutions 
  of 
  this 
  

   subgroup 
  can 
  transform 
  all 
  the 
  substitutions 
  of 
  R 
  in 
  the 
  

   same 
  manner. 
  From 
  the 
  preceding 
  paragraph 
  it 
  follows 
  

   that 
  this 
  subgroup 
  transforms 
  R 
  into 
  itself 
  in 
  every 
  possible 
  

   manner. 
  

  

  Since 
  the 
  order 
  of 
  the 
  given 
  subgroup 
  is 
  equal 
  to 
  the 
  

   order 
  of 
  the 
  entire 
  group 
  divided 
  by 
  n, 
  the 
  latter 
  must 
  

   contain 
  just 
  n 
  substitutions 
  that 
  are 
  commutative 
  to 
  every 
  

   substitution 
  of 
  R. 
  These 
  form 
  a 
  regular 
  group, 
  which 
  coin- 
  

   cides 
  with 
  R 
  only 
  when 
  R 
  is 
  commutative. 
  Each 
  one 
  of 
  

   these 
  two 
  regular 
  groups 
  is 
  said 
  to 
  be 
  the 
  associate* 
  of 
  the 
  

   other. 
  Before 
  proceeding 
  further 
  in 
  the 
  consideration 
  of 
  

   the 
  simple 
  isomorphisms 
  of 
  R 
  to 
  itself, 
  it 
  seems 
  well 
  to 
  give 
  

   some 
  definitions 
  which 
  apply 
  to 
  the 
  general 
  substitution- 
  

   group. 
  

  

  § 
  1. 
  Definitions 
  and 
  General 
  Considerations. 
  

  

  If 
  we 
  regard 
  the 
  substitutions 
  of 
  G 
  as 
  elements, 
  we 
  observe 
  

   that 
  a 
  substitution 
  of 
  these 
  elements 
  corresponds 
  to 
  every 
  

   simple 
  isomorphism 
  of 
  G 
  to 
  itself 
  and 
  a 
  substitution-group 
  to 
  

   all 
  the 
  possible 
  isomorphisms. 
  This 
  substitution-group 
  (X) 
  

   has 
  been 
  called 
  the 
  group 
  of 
  isomorphisms 
  of 
  G. 
  Its 
  degree 
  

   is 
  the 
  order 
  of 
  G 
  diminished 
  by 
  the 
  number 
  of 
  its 
  substi- 
  

   tutions 
  that 
  correspond 
  to 
  themselves 
  in 
  every 
  simple 
  iso- 
  

  

  * 
  •' 
  Quarterly 
  Journal 
  of 
  Mathematics/ 
  vol. 
  xxviii. 
  p. 
  249. 
  The 
  pre- 
  

   ceding 
  seems 
  to 
  be 
  an 
  easy 
  proof 
  of 
  Jordan's 
  theorem 
  in 
  regard 
  to 
  the 
  

   number 
  of 
  substitutions 
  that 
  are 
  commutative 
  to 
  every 
  substitution 
  of 
  a 
  

   given 
  regular 
  group. 
  Cf. 
  Traite 
  des 
  Substitutions, 
  p. 
  60. 
  

  

  