﻿234 
  Dr. 
  G. 
  A. 
  Miller 
  on 
  the 
  Simple 
  Isomorphisms 
  

  

  temperature, 
  and 
  inferred 
  that 
  there 
  was 
  a 
  correction 
  needed 
  

   involving 
  the 
  first 
  power 
  of 
  the 
  cooling-effect. 
  Both 
  his 
  

   investigation 
  and 
  my 
  own 
  involve 
  the 
  assumption 
  that 
  an 
  

   empirical 
  formula 
  found 
  to 
  hold 
  through 
  a 
  short 
  range 
  of 
  

   temperature 
  can 
  he 
  used 
  for 
  any 
  temperature 
  however 
  high, 
  

   and 
  hence 
  neither 
  his 
  result 
  nor 
  mine 
  is 
  conclusively 
  estab- 
  

   lished 
  ; 
  but 
  it 
  seems 
  interesting 
  to 
  show 
  that 
  the 
  employment 
  

   of 
  a 
  new 
  expression, 
  at 
  least 
  as 
  good 
  as 
  Lord 
  Kelvin's, 
  for 
  

   the 
  cooling-effect, 
  leads 
  to 
  a 
  smaller 
  value 
  for 
  the 
  correction. 
  

  

  XXY1. 
  On 
  the 
  Simple 
  Isomorphisms 
  of 
  a 
  Substitution- 
  Group 
  

   to 
  itself. 
  Bij 
  G. 
  A. 
  Miller, 
  Ph.D.* 
  

  

  MOST 
  of 
  the 
  difficulties 
  connected 
  with 
  the 
  study 
  of 
  

   groups 
  reside 
  either 
  in 
  the 
  simple 
  groups 
  or 
  in 
  the 
  

   general 
  problem 
  of 
  isomorphisms. 
  One 
  of 
  the 
  fundamental 
  

   elements 
  of 
  this 
  problem 
  consists 
  of 
  the 
  simple 
  isomorphisms 
  

   of 
  a 
  group 
  to 
  itself. 
  It 
  is 
  our 
  object 
  to 
  give 
  a 
  general 
  outline 
  

   of 
  this 
  element, 
  together 
  with 
  a 
  few 
  details 
  which 
  appear 
  new 
  

   and 
  of 
  considerable 
  importance 
  in 
  the 
  study 
  of 
  the 
  intransitive 
  

   substitution-groups. 
  

  

  The 
  simplest 
  method 
  of 
  making 
  a 
  group 
  (G) 
  simply 
  iso- 
  

   morphic 
  to 
  itself 
  is 
  that 
  by 
  which 
  we 
  write 
  after 
  each 
  one 
  of 
  

   its 
  substitutions 
  f 
  the 
  transform 
  with 
  respect 
  to 
  some 
  substi- 
  

   tution 
  that 
  is 
  commutative 
  to 
  G. 
  That 
  we 
  obtain 
  a 
  simple 
  

   isomorphism 
  in 
  this 
  way, 
  follows 
  directly 
  from 
  the 
  equation 
  

  

  s~ 
  1 
  t 
  1 
  s 
  . 
  s~ 
  1 
  t 
  2 
  s 
  = 
  s~H 
  1 
  t 
  2 
  s. 
  

  

  From 
  this 
  equation 
  it 
  also 
  follows 
  that 
  we 
  oblain 
  a 
  simple 
  

   isomorphism 
  when 
  the 
  transforming 
  substitution 
  is 
  not 
  com- 
  

   mutative 
  to 
  G. 
  In 
  this 
  case 
  it 
  would, 
  however, 
  not 
  be 
  a 
  

   simple 
  isomorphism 
  of 
  G 
  to 
  itself. 
  

  

  Instead 
  of 
  transforming 
  every 
  substitution 
  of 
  G 
  by 
  the 
  

   same 
  substitution, 
  w 
  T 
  e 
  may 
  employ 
  different 
  transformers 
  for 
  

   the 
  different 
  substitutions. 
  It 
  is 
  necessary 
  and 
  sufficient 
  that 
  

   all 
  such 
  transformers 
  have 
  the 
  same 
  effect 
  upon 
  the 
  sub- 
  

   stitutions 
  to 
  which 
  they 
  are 
  applied, 
  regarded 
  as 
  operators, 
  as 
  

   a 
  given 
  transforming 
  operator 
  has 
  upon 
  the 
  corresponding 
  

   operators 
  of 
  the 
  simply 
  isomorphic 
  operation-group. 
  

  

  Suppose 
  that 
  a 
  regular 
  group 
  (li) 
  is 
  made 
  simply 
  iso- 
  

   morphic 
  to 
  itself 
  in 
  any 
  one 
  of 
  the 
  possible 
  ways. 
  We 
  may 
  

   suppose 
  that 
  all 
  the 
  substitutions, 
  except 
  identity, 
  begin 
  with 
  

  

  * 
  Communicated 
  by 
  the 
  Author 
  ; 
  having 
  been 
  read 
  at 
  the 
  Detroit 
  

   Meeting 
  of 
  the 
  American 
  Association 
  for 
  the 
  Advancement 
  of 
  Science, 
  

   1897. 
  

  

  t 
  Unless 
  the 
  contrary 
  is 
  stated, 
  the 
  groups 
  under 
  consideration 
  are 
  

   supposed 
  to 
  be 
  general 
  substitution-groups. 
  

  

  