﻿of 
  a 
  Substitution- 
  Group 
  to 
  itself. 
  241 
  

  

  of 
  isomorphisms 
  with 
  respect 
  to 
  the 
  arrangement 
  of 
  its 
  sub- 
  

   stitutions 
  in 
  the 
  given 
  form 
  is 
  of 
  great 
  importance 
  in 
  the 
  

   study 
  of 
  simple 
  isomorphisms. 
  For 
  instance, 
  if 
  the 
  group 
  of 
  

   isomorphisms 
  of 
  G 
  is 
  the 
  symmetric 
  group 
  of 
  order 
  6, 
  and 
  

   its 
  subgroup 
  that 
  corresponds 
  to 
  the 
  transforms 
  of 
  the 
  iden- 
  

   tical 
  is 
  of 
  order 
  2 
  or 
  3 
  ? 
  then 
  will 
  G 
  have 
  just 
  two 
  distinct 
  

   simple 
  isomorphisms 
  to 
  itself. 
  This 
  is 
  evident 
  when 
  the 
  

   subgroup 
  is 
  of 
  order 
  3. 
  When 
  it 
  is 
  of 
  order 
  2 
  we 
  may 
  

   arrange 
  the 
  substitutions 
  of 
  the 
  symmetric 
  group 
  as 
  follows 
  : 
  

  

  5,, 
  s 
  2 
  = 
  1 
  ab 
  

  

  ts^ 
  ts 
  2 
  = 
  ac 
  acb 
  

  

  s^t^si, 
  s~ 
  l 
  t~ 
  1 
  s 
  2 
  = 
  ctbc 
  be 
  

  

  It 
  is 
  clear 
  that 
  we 
  should 
  have 
  arrived 
  at 
  the 
  same 
  result 
  

   by 
  using 
  either 
  one 
  of 
  the 
  other 
  two 
  subgroups 
  of 
  order 
  2, 
  

   since 
  these 
  three 
  subgroups 
  are 
  conjugate. 
  When 
  G 
  is 
  

   transitive 
  the 
  problem 
  of 
  finding 
  the 
  number 
  of 
  its 
  distinct 
  

   simple 
  isomorphisms 
  is 
  thus 
  reduced 
  to 
  the 
  following 
  two 
  

   more 
  elementary 
  problems 
  : 
  — 
  (1) 
  To 
  determine 
  the 
  subgroup 
  

   of 
  its 
  I 
  which 
  corresponds 
  to 
  all 
  the 
  transforms 
  of 
  the 
  iden- 
  

   tical 
  isomorphism 
  ; 
  and 
  (2) 
  To 
  arrange 
  the 
  substitutions 
  of 
  I 
  

   in 
  the 
  given 
  form. 
  When 
  G 
  is 
  intransitive 
  the 
  number 
  of 
  

   distinct 
  simple 
  isomorphisms 
  cannot 
  exceed 
  the 
  number 
  found 
  

   in 
  this 
  way. 
  

  

  Tn 
  making 
  G 
  simply 
  isomorphic 
  to 
  itself 
  it 
  is 
  frequently 
  

   convenient 
  to 
  inquire 
  whether 
  a 
  given 
  subgroup 
  may 
  corre- 
  

   spond 
  to 
  another 
  given 
  subgroup. 
  We 
  have 
  observed 
  that 
  

   such 
  subgroups 
  must 
  be 
  of 
  the 
  same 
  type. 
  Suppose 
  that 
  this 
  

   condition 
  is 
  satisfied, 
  and 
  that 
  the 
  corresponding 
  substitutions 
  

   are, 
  in 
  order, 
  

  

  5 
  lj 
  s 
  2) 
  s 
  3y 
  • 
  • 
  V 
  S 
  \J 
  5 
  1 
  y 
  S 
  2 
  ) 
  s 
  3 
  > 
  • 
  • 
  -j 
  S 
  \ 
  • 
  

  

  Let 
  t 
  x 
  and 
  t 
  2 
  be 
  two 
  substitutions 
  of 
  G 
  such 
  that 
  (1) 
  all 
  the 
  

   substitutions 
  of 
  two 
  larger 
  subgroups 
  of 
  G 
  (or 
  of 
  G 
  itself) 
  are 
  

   of 
  the 
  form 
  

  

  *?*0 
  *W 
  ("=1, 
  2, 
  ...,: 
  /3=1,2, 
  ...,\) 
  ; 
  

  

  (2) 
  the 
  first 
  power 
  of 
  t 
  ± 
  that 
  occurs 
  among 
  the 
  s's 
  corresponds 
  

   to 
  the 
  same 
  first 
  power 
  of 
  £ 
  2 
  that 
  occurs 
  in 
  the 
  given 
  simple 
  

   isomorphic 
  subgroups 
  ; 
  and 
  (3) 
  the 
  same 
  powers 
  of 
  t 
  x 
  and 
  t 
  2 
  

   transform 
  the 
  corresponding 
  generating 
  substitutions 
  of 
  the 
  

   given 
  subgroups 
  into 
  the 
  same 
  power 
  of 
  the 
  t'& 
  multiplied 
  

   into 
  corresponding 
  s's 
  : 
  then 
  will 
  these 
  larger 
  subgroups 
  (or 
  

   G) 
  be 
  also 
  simply 
  isomorphic. 
  

  

  