﻿238 
  Dr. 
  Gr. 
  A. 
  Miller 
  on 
  the 
  Simple 
  Isomorphisms 
  

  

  degree 
  n 
  which 
  transforms 
  R 
  into 
  all 
  its 
  simple 
  isomorphisms 
  

   to 
  itself 
  cannot 
  be 
  less 
  than 
  ?i-4-2. 
  In 
  other 
  words, 
  a 
  substi- 
  

   tution 
  which 
  transforms 
  a 
  regular 
  group 
  of 
  degree 
  n 
  into 
  

   itself 
  must 
  be 
  of 
  degree 
  ~>n-±-2, 
  if 
  it 
  is 
  not 
  identity. 
  We 
  

   proceed 
  to 
  determine 
  this 
  degree 
  somewhat 
  more 
  accurately. 
  

   Suppose 
  that 
  m 
  is 
  the 
  order 
  of 
  the 
  largest 
  maximal 
  sub- 
  

   group 
  of 
  R, 
  and 
  that 
  some 
  substitution 
  that 
  is 
  commutative 
  

   to 
  every 
  substitution 
  of 
  this 
  subgroup 
  transforms 
  R 
  into 
  itself; 
  

   then 
  will 
  the 
  class 
  of 
  the 
  group 
  that 
  transforms 
  R 
  into 
  all 
  its 
  

   simple 
  isomorphisms 
  to 
  itself 
  be 
  n 
  — 
  m. 
  This 
  number 
  cannot 
  

  

  Yi 
  (2) 
  1) 
  

  

  be 
  less 
  than 
  — 
  — 
  ,p 
  being 
  the 
  smallest 
  prime 
  number 
  which 
  

  

  is 
  a 
  factor 
  of 
  the 
  order 
  of 
  R. 
  When 
  p 
  = 
  2, 
  we 
  have 
  the 
  limit 
  

   at 
  which 
  we 
  arrived 
  in 
  the 
  preceding 
  paragraph. 
  

  

  The 
  largest 
  group 
  of 
  degree 
  n 
  that 
  transforms 
  R 
  into 
  all 
  

   its 
  simple 
  isomorphisms 
  to 
  itself 
  has 
  a 
  n, 
  1 
  isomorphism 
  to 
  the 
  

   I 
  of 
  R. 
  To 
  identity 
  in 
  I 
  we 
  may 
  let 
  either 
  R 
  or 
  its 
  associate 
  

   correspond. 
  The 
  average 
  degree 
  of 
  the 
  n 
  substitutions 
  which 
  

   correspond 
  to 
  any 
  substitution 
  of 
  I 
  is 
  n 
  — 
  1. 
  Hence, 
  some 
  of 
  

   these 
  must 
  be 
  of 
  degree 
  n 
  } 
  in 
  case 
  at 
  least 
  one 
  of 
  them 
  is 
  of 
  a 
  

   lower 
  degree 
  than 
  n— 
  1. 
  

  

  When 
  R 
  contains 
  no 
  substitution 
  besides 
  identity 
  that 
  is 
  

   commutative 
  to 
  all 
  its 
  substitutions, 
  it 
  and 
  its 
  associate 
  have 
  

   only 
  one 
  common 
  substitution. 
  The 
  largest 
  group 
  of 
  

   decree 
  n 
  which 
  transforms 
  R 
  into 
  all 
  its 
  cooredient 
  iso- 
  

   morphisms 
  is 
  of 
  order 
  n 
  2 
  . 
  Its 
  subgroup 
  which 
  contains 
  all 
  

   its 
  substitutions 
  that 
  do 
  not 
  involve 
  a 
  given 
  element, 
  is 
  

   simply 
  isomorphic 
  to 
  R 
  as 
  well 
  as 
  to 
  its 
  associate. 
  Hence 
  

   at 
  least 
  such 
  an 
  R 
  must 
  be 
  the 
  transform 
  of 
  its 
  associate. 
  

  

  In 
  general, 
  the 
  substitutions 
  that 
  are 
  common 
  to 
  R 
  and 
  

   its 
  associate 
  form 
  a 
  characteristic 
  subgroup 
  of 
  both. 
  The 
  

   quotient 
  group 
  of 
  R 
  and 
  its 
  associate 
  with 
  respect 
  to 
  this 
  

   subgroup 
  must 
  be 
  of 
  a 
  composite 
  order. 
  The 
  largest 
  group 
  

   of 
  degree 
  n 
  that 
  transforms 
  R 
  only 
  according 
  to 
  its 
  cogredient 
  

   isomorphisms 
  is 
  of 
  order 
  n^-r-m, 
  m 
  being 
  the 
  order 
  of 
  the 
  

   given 
  characteristic 
  subgroup. 
  The 
  subgroup 
  of 
  order 
  n-^-m 
  

   which 
  contains 
  all 
  the 
  substitutions 
  of 
  this 
  group 
  that 
  do 
  not 
  

   involve 
  a 
  given 
  element 
  is 
  the 
  group 
  of 
  cogredient 
  iso- 
  

   morphisms 
  of 
  R 
  and 
  its 
  associate. 
  When 
  ?n 
  = 
  n, 
  R 
  is 
  com- 
  

   mutative, 
  and 
  vice 
  versa. 
  

  

  By 
  multiplying 
  all 
  the 
  substitutions 
  of 
  R, 
  first 
  on 
  the 
  rio-ht 
  

   and 
  then 
  on 
  the 
  left, 
  by 
  each 
  one 
  of 
  them 
  we 
  obtain 
  two 
  

   squares 
  containing 
  n 
  2 
  substitutions, 
  each 
  substitution 
  occurring 
  

   n 
  times 
  in 
  each 
  square. 
  The 
  substitutions 
  by 
  means 
  of 
  which 
  

   we 
  obtain 
  all 
  the 
  lines 
  of 
  a 
  square 
  from 
  any 
  one 
  of 
  them 
  form 
  

  

  