﻿240 
  Dr. 
  G. 
  A. 
  Miller 
  on 
  the 
  Simple 
  Isomorphisms 
  

  

  isomorphism 
  represented 
  by 
  any 
  transform 
  of 
  the 
  identical 
  

   form 
  a 
  subgroup 
  of 
  I. 
  We 
  shall 
  represent 
  the 
  substitutions 
  

   of 
  this 
  subgroup 
  by 
  s 
  1? 
  s 
  2 
  , 
  s 
  3 
  , 
  . 
  . 
  . 
  s 
  y 
  

  

  Let 
  i 
  be 
  a 
  substitution 
  of 
  1 
  that 
  belongs 
  to 
  any 
  other 
  

   distinct 
  isomorphism. 
  Then 
  will 
  

  

  ISi) 
  IS2 
  , 
  tSfy 
  . 
  . 
  . 
  ts^ 
  

  

  belong 
  to 
  the 
  same 
  distinct 
  isomorphism. 
  Since 
  the 
  inverse 
  

   of 
  each 
  one 
  of 
  these 
  corresponds 
  to 
  the 
  interchange 
  of 
  the 
  two 
  

   constituents 
  in 
  the 
  corresponding 
  isomorphism 
  it 
  must 
  belong 
  

   to 
  the 
  same 
  distinct 
  isomorphism. 
  If 
  one 
  of 
  these 
  {ts 
  a 
  )~ 
  l 
  is 
  

   not 
  included 
  in 
  the 
  given 
  line, 
  the 
  following 
  substitutions 
  of 
  

   I 
  must 
  belong 
  to 
  the 
  same 
  distinct 
  isomorphism, 
  

  

  s~ 
  1 
  t~ 
  1 
  s 
  1 
  , 
  S~ 
  l 
  t~ 
  1 
  So, 
  s~ 
  1 
  t~ 
  1 
  s 
  3 
  . 
  . 
  . 
  ., 
  5 
  _1 
  i 
  _1 
  6\. 
  

  

  a 
  1 
  ' 
  a 
  *~ 
  a 
  °' 
  "a 
  A 
  

  

  All 
  of 
  these 
  are 
  evidently 
  different 
  from 
  the 
  preceding 
  sub- 
  

   stitutions. 
  If 
  these 
  two 
  lines 
  do 
  not 
  contain 
  all 
  the 
  inverses 
  

   of 
  their 
  2\ 
  substitutions, 
  we 
  add 
  a 
  new 
  line 
  by 
  multiplying 
  

   such 
  an 
  inverse 
  into 
  the 
  given 
  s's, 
  &c. 
  

  

  Since 
  all 
  the 
  substitutions 
  of 
  I 
  that 
  may 
  be 
  obtained 
  in 
  

   this 
  manner 
  are 
  of 
  the 
  form 
  

  

  s„<v 
  S{ 
  0,8=1,2, 
  ...,X: 
  y 
  =l. 
  -1) 
  

  

  their 
  number 
  cannot 
  exceed 
  2A 
  2 
  . 
  It 
  is 
  easy 
  to 
  verify 
  that 
  

   this 
  maximum 
  is 
  reached 
  in 
  the 
  group 
  of 
  order 
  168 
  and 
  

   degree 
  7 
  if 
  we 
  take 
  for 
  the 
  s's 
  its 
  subgroup 
  of 
  order 
  6. 
  This 
  

   is 
  the 
  group 
  of 
  isomorphisms 
  of 
  the 
  group 
  of 
  order 
  8 
  which 
  

   contains 
  no 
  substitution 
  whose 
  order 
  exceeds 
  2*. 
  

  

  An 
  isomorphism 
  may 
  be 
  transformed 
  into 
  another 
  with 
  

   the 
  same 
  constituent 
  groups 
  (1) 
  by 
  transforming 
  either 
  or 
  

   both 
  of 
  its 
  constituents, 
  (2) 
  by 
  interchanging 
  the 
  constituents, 
  

   and 
  (3) 
  by 
  transforming 
  a 
  part 
  of 
  one 
  constituent 
  into 
  a 
  part 
  

   of 
  the 
  other. 
  All 
  the 
  isomorphisms 
  which 
  may 
  be 
  obtained 
  

   by 
  the 
  first 
  two 
  methods 
  or 
  their 
  combination 
  correspond 
  to 
  

   substitutions 
  of 
  I 
  that 
  are 
  included 
  in 
  the 
  given 
  rectangular 
  

   form. 
  As 
  the 
  last 
  method 
  does 
  not 
  apply 
  to 
  transitive 
  groups 
  

   we 
  observe 
  that 
  the 
  given 
  form 
  includes 
  all 
  the 
  substitutions 
  

   of 
  I 
  which 
  correspond 
  to 
  a 
  given 
  distinct 
  isomorphism 
  when 
  

   G 
  is 
  transitive. 
  When 
  G 
  is 
  intransitive, 
  several 
  of 
  these 
  

   rectangular 
  forms 
  may 
  belong 
  to 
  the 
  same 
  distinct 
  iso- 
  

   morphism. 
  

  

  From 
  what 
  precedes 
  it 
  is 
  evident 
  that 
  a 
  study 
  of 
  the 
  group 
  

  

  * 
  Cf. 
  Moore, 
  Bulletin 
  of 
  the 
  American 
  Mathematical 
  Society 
  (1894) 
  

   vol. 
  i. 
  p. 
  61. 
  

  

  