﻿of 
  a 
  Substitution- 
  Group 
  to 
  itself. 
  239 
  

  

  a 
  group 
  which 
  is 
  simply 
  isomorphic 
  to 
  R. 
  Each 
  of 
  these 
  

   substitutions 
  of 
  one 
  square 
  is 
  commutative 
  to 
  every 
  one 
  of 
  

   those 
  of 
  the 
  other 
  square. 
  If 
  a 
  substitution 
  is 
  commutative 
  

   to 
  every 
  substitution 
  of 
  R, 
  it 
  must 
  clearly 
  be 
  of 
  degree 
  n. 
  

   This 
  proof 
  of 
  the 
  given 
  theorem 
  of 
  Jordan 
  is 
  due 
  to 
  Frattini. 
  

   It 
  follows 
  directly 
  from 
  this 
  method 
  of 
  proof 
  that 
  every 
  R 
  is 
  

   the 
  transform 
  of 
  its 
  associate. 
  

  

  Since 
  all 
  the 
  simple 
  isomorphisms 
  of 
  R 
  to 
  itself 
  can 
  be 
  

   obtained 
  by 
  transforming 
  the 
  identical 
  isomorphism, 
  we 
  need 
  

   to 
  transform 
  only 
  one 
  of 
  the 
  two 
  constituent 
  groups 
  in 
  any 
  

   one 
  such 
  isomorphism 
  in 
  order 
  to 
  obtain 
  all 
  the 
  others. 
  We 
  

   shall 
  see 
  in 
  the 
  next 
  section 
  that 
  it 
  is 
  not 
  generally 
  possible 
  

   to 
  obtain 
  all 
  the 
  transforms 
  of 
  a 
  given 
  isomorphism 
  of 
  a 
  general 
  

   substitution-group 
  in 
  this 
  manner. 
  

  

  § 
  3. 
  Simple 
  Isomorphisms 
  of 
  G 
  to 
  itself, 
  

   -Let 
  Si, 
  s 
  2 
  > 
  $$, 
  . 
  . 
  . 
  s 
  m 
  

  

  be 
  m 
  substitutions 
  of 
  order 
  two 
  such 
  that 
  no 
  two 
  of 
  them 
  

   contain 
  any 
  common 
  element, 
  and 
  that 
  the 
  degree 
  of 
  each 
  

   exceeds 
  the 
  sum 
  of 
  the 
  degrees 
  of 
  all 
  those 
  which 
  precede 
  it. 
  

   The 
  group 
  of 
  order 
  2 
  a 
  which 
  is 
  generated 
  by 
  any 
  a 
  of 
  these 
  

   substitutions 
  contains 
  no 
  two 
  substitutions 
  of 
  the 
  same 
  degree. 
  

   This 
  group 
  has 
  

  

  (2 
  a 
  -l)(2 
  a 
  -2)(2 
  a 
  -2 
  2 
  )(2 
  a 
  -2 
  3 
  ) 
  . 
  . 
  . 
  (2 
  a 
  -2 
  a 
  ~ 
  1 
  ) 
  

  

  simple 
  isomorphisms 
  to 
  itself. 
  None 
  of 
  these 
  except 
  the 
  

   identical 
  can 
  be 
  obtained 
  by 
  transforming 
  the 
  identical. 
  The 
  

   transforms 
  of 
  the 
  others 
  will, 
  in 
  general, 
  lead 
  to 
  different 
  

   isomorphisms. 
  

  

  The 
  G 
  which 
  we 
  have 
  just 
  given 
  is 
  an 
  extreme 
  case. 
  It 
  is 
  

   generally 
  possible 
  to 
  obtain 
  a 
  number 
  of 
  different 
  iso- 
  

   morphisms 
  by 
  transforming 
  the 
  identical. 
  If 
  the 
  G 
  which 
  

   is 
  made 
  simple 
  isomorphic 
  to 
  itself 
  is 
  transitive, 
  and 
  its 
  order 
  

   exceeds 
  2, 
  this 
  is 
  always 
  possible. 
  For 
  when 
  it 
  is 
  not 
  com- 
  

   mutative, 
  we 
  may 
  obtain 
  such 
  isomorphisms 
  by 
  transforming 
  

   it 
  by 
  its 
  own 
  substitutions. 
  When 
  it 
  is 
  commutative 
  it 
  must 
  

   be 
  regular 
  and 
  must 
  have 
  different 
  simple 
  isomorphisms 
  to 
  

   itself 
  unless 
  it 
  has 
  only 
  one 
  substitution 
  of 
  a 
  given 
  order. 
  In 
  

   this 
  case 
  it 
  must 
  be 
  the 
  transitive 
  group 
  of 
  order 
  2. 
  

  

  When 
  the 
  transforms 
  of 
  the 
  identical 
  isomorphism 
  of 
  G 
  do 
  

   not 
  give 
  the 
  total 
  number 
  of 
  its 
  simple 
  isomorphisms 
  to 
  

   itself, 
  G 
  has 
  more 
  than 
  one 
  distinct 
  isomorphism. 
  It 
  is 
  

   important 
  to 
  find 
  the 
  numbers 
  of 
  the 
  substitutions 
  of 
  I 
  which 
  

   belong 
  to 
  each 
  distinct 
  isomorphism 
  of 
  G. 
  We 
  have 
  ob- 
  

   served 
  that 
  the 
  substitutions 
  of 
  I 
  which 
  belong 
  to 
  the 
  distinct 
  

  

  S 
  2 
  

  

  