﻿242 
  Simple 
  Isomorphisms 
  of 
  a 
  Substitution- 
  Group 
  to 
  itself. 
  

  

  The 
  proof 
  of 
  this 
  statement 
  is 
  evident 
  if 
  we 
  arrange 
  the 
  

   substitutions 
  as 
  follows 
  : 
  — 
  

  

  s 
  1 
  s 
  2 
  s 
  3 
  - 
  

   *i^i 
  '1^2 
  '1^3 
  

   1 
  ^1 
  *i 
  ^2 
  1 
  ^3 
  

  

  h* 
  s 
  i 
  h* 
  s 
  2 
  h* 
  8 
  : 
  

  

  . 
  . 
  ^s 
  A 
  

  

  *l' 
  

  

  < 
  s 
  2 
  

  

  *J 
  • 
  

  

  

  W 
  

  

  *2^2 
  

  

  t 
  2 
  s 
  3 
  . 
  

  

  • 
  • 
  t>2 
  s 
  k 
  

  

  t 
  *« 
  ' 
  

  

  ^2 
  ^2 
  

  

  h 
  5 
  3 
  • 
  

  

  t 
  2 
  C 
  ' 
  

  

  ■ 
  ^'a 
  hW 
  fety 
  *W 
  • 
  • 
  • 
  W 
  

  

  ^? 
  + 
  ? 
  ^2 
  +1 
  being 
  the 
  first 
  power 
  of 
  the 
  t's 
  that 
  is 
  found 
  in 
  the 
  

   first 
  row. 
  Under 
  the 
  given 
  conditions 
  the 
  products 
  of 
  

   corresponding 
  substitutions 
  in 
  the 
  two 
  rectangles 
  must 
  

   correspond. 
  

  

  § 
  4. 
  The 
  Group 
  I. 
  

  

  If 
  m 
  is 
  the 
  number 
  of 
  substitutions 
  of 
  G 
  that 
  correspond 
  

   to 
  themselves 
  in 
  all 
  the 
  simple 
  isomorphisms 
  of 
  G 
  to 
  itself, 
  

   the 
  degree 
  of 
  I 
  is 
  g 
  — 
  m, 
  g 
  being 
  the 
  order 
  of 
  G. 
  It 
  is 
  

   evident 
  that 
  mj;l. 
  Professor 
  Moore 
  has 
  examined 
  the 
  case 
  

   when 
  I 
  is 
  transitive 
  and 
  of 
  degree 
  g 
  — 
  1 
  *", 
  arriving 
  at 
  some 
  

   very 
  interesting 
  results. 
  As 
  a 
  rule 
  I 
  is 
  intransitive. 
  In- 
  

   stead 
  of 
  considering 
  the 
  entire 
  group 
  it 
  is 
  frequently 
  con- 
  

   venient 
  to 
  consider 
  a 
  constituent 
  which 
  is 
  simply 
  isomorphic 
  

   to 
  it. 
  As 
  elements 
  of 
  such 
  a 
  constituent 
  we 
  may 
  take 
  those 
  

   which 
  correspond 
  to 
  any 
  system 
  of 
  generators 
  of 
  G 
  and 
  to 
  

   the 
  substitutions 
  to 
  which 
  these 
  generators 
  correspond 
  in 
  any 
  

   simple 
  isomorphism 
  of 
  G 
  to 
  itself. 
  

  

  I 
  has 
  alwaj^s 
  a 
  1 
  , 
  a 
  isomorphism 
  to 
  the 
  largest 
  group 
  of 
  

   degree 
  n 
  that 
  transforms 
  G 
  into 
  itself, 
  n 
  being 
  the 
  degree 
  of 
  

   G. 
  We 
  have 
  seen 
  that 
  a 
  = 
  n 
  when 
  G 
  is 
  regular. 
  When 
  G 
  

   is 
  a 
  non-regular 
  transitive 
  group 
  the 
  substitutions 
  that 
  are 
  

   commutative 
  to 
  each 
  of 
  its 
  substitutions 
  must 
  be 
  of 
  degree 
  

   or 
  n. 
  Since 
  they 
  form 
  a 
  group 
  their 
  number 
  cannot 
  exceed 
  

   n. 
  As 
  every 
  substitution 
  of 
  this 
  group 
  is 
  commutative 
  to 
  

   substitutions 
  whose 
  degree 
  is 
  less 
  than 
  n 
  it 
  must 
  be 
  intransitive. 
  

   Hence 
  a 
  cannot 
  exceed 
  n-f-2 
  for 
  such 
  a 
  G. 
  If 
  G 
  contains 
  a 
  

   subgroup 
  of 
  degree 
  n— 
  1, 
  a 
  is 
  evidently 
  1. 
  When 
  I 
  is 
  simply 
  

   isomorphic 
  to 
  G 
  it 
  is 
  said 
  to 
  be 
  a 
  complete 
  group 
  t> 
  and 
  

   vice 
  versa. 
  

  

  Paris, 
  July 
  1897. 
  

  

  * 
  Moore, 
  Bulletin 
  of 
  the 
  American 
  Mathematical 
  Society 
  (1895), 
  vol. 
  ii. 
  

   p. 
  33. 
  

  

  t 
  Holder, 
  Mathematische 
  Annalen, 
  -vol. 
  xlvi. 
  p. 
  325, 
  

  

  