﻿of 
  a 
  Substitution- 
  Group 
  to 
  itself, 
  237 
  

  

  gredient 
  simple 
  isomorphisms 
  to 
  itself, 
  these 
  six 
  types 
  of 
  

   subgroups 
  reduce 
  to 
  two, 
  since 
  the 
  characteristic, 
  self- 
  

   transform, 
  and 
  self-conjugate 
  subgroups 
  become 
  identical, 
  as 
  

   well 
  as 
  the 
  isomorphic, 
  transform, 
  and 
  conjugate 
  subgroups. 
  

   With 
  respect 
  to 
  groups 
  like 
  the 
  operation 
  or 
  abstract 
  groups 
  

   and 
  the 
  regular 
  substitution-groups, 
  which 
  admit 
  only 
  trans- 
  

   form 
  simple 
  isomorphisms 
  to 
  themselves, 
  they 
  reduce 
  to 
  four 
  

   types, 
  since 
  the 
  characteristic 
  subgroups 
  and 
  the 
  self-transform 
  

   become 
  identical, 
  as 
  well 
  as 
  the 
  isomorphic 
  and 
  the 
  transform. 
  

  

  There 
  is 
  a 
  special 
  type 
  of 
  characteristic 
  subgroups 
  to 
  which 
  

   attention 
  should 
  be 
  called, 
  viz. 
  that 
  formed 
  by 
  all 
  the 
  commu- 
  

   tators 
  of 
  G. 
  It 
  has 
  been 
  called 
  by 
  Dedekind 
  the 
  commutator 
  

   subgroup 
  of 
  G. 
  As 
  Gr 
  has 
  only 
  one 
  such 
  subgroup, 
  it 
  must 
  

   correspond 
  to 
  itself 
  in 
  all 
  the 
  simple 
  isomorphisms 
  of 
  G 
  

   to 
  itself. 
  In 
  fact, 
  a 
  subgroup 
  belonging 
  to 
  any 
  one 
  of 
  the 
  

   seven 
  types 
  that 
  have 
  been 
  defined 
  must 
  correspond 
  to 
  one 
  of 
  

   the 
  same 
  type 
  in 
  every 
  simple 
  isomorphism. 
  

  

  As 
  the 
  cogredient 
  isomorphisms 
  correspond 
  to 
  a 
  self- 
  

   conjugate 
  subgroup 
  of 
  I, 
  this 
  group 
  must 
  have 
  an 
  a, 
  1 
  

   isomorphism 
  to 
  some 
  other 
  group 
  (I'), 
  ex. 
  being 
  the 
  order 
  of 
  

   the 
  given 
  self-conjugate 
  subgroup 
  of 
  I. 
  Those 
  a 
  isomorphisms 
  

   which 
  correspond 
  to 
  the 
  same 
  operator 
  of 
  1' 
  are 
  said 
  to 
  be 
  of 
  

   the 
  same 
  class 
  *. 
  All 
  of 
  them 
  may 
  be 
  obtained 
  from 
  any 
  

   one 
  by 
  transforming 
  it 
  with 
  respect 
  to 
  substitutions 
  of 
  G 
  ; 
  

   for 
  the 
  substitution 
  of 
  I 
  which 
  corresponds 
  to 
  the 
  transform 
  

   of 
  a 
  given 
  isomorphism 
  is 
  obtained 
  by 
  multiplying 
  the 
  sub- 
  

   stitution 
  which 
  corresponds 
  to 
  the 
  isomorphism 
  into 
  that 
  

   which 
  corresponds 
  to 
  the 
  transforming 
  operator. 
  

  

  § 
  2. 
  Simple 
  Isomorphisms 
  of 
  R 
  to 
  itself 
  

  

  Suppose 
  that 
  all 
  the 
  substitutions 
  of 
  R 
  except 
  identity 
  begin 
  

   with 
  the 
  same 
  element 
  (%), 
  and 
  that 
  each 
  one 
  of 
  them 
  is 
  

   denoted 
  by 
  its 
  second 
  element. 
  No 
  two 
  substitutions 
  will 
  

   thus 
  be 
  denoted 
  by 
  the 
  same 
  element. 
  Each 
  substitution 
  of 
  

   the 
  subgroup, 
  which 
  contains 
  all 
  the 
  substitutions 
  that 
  do 
  not 
  

   involve 
  a 
  iy 
  of 
  the 
  largest 
  group 
  of 
  degree 
  n 
  that 
  transforms 
  R 
  

   into 
  itself 
  transforms 
  the 
  substitutions 
  of 
  R 
  in 
  exactly 
  the 
  

   same 
  manner 
  as 
  its 
  own 
  elements. 
  Hence 
  this 
  subgroup 
  may 
  

   be 
  considered 
  the 
  group 
  of 
  isomorphisms 
  of 
  R. 
  By 
  changing 
  

   the 
  notation 
  of 
  the 
  substitutions 
  of 
  R 
  we 
  may 
  obtain 
  any 
  one 
  

   of 
  the 
  conjugates 
  of 
  this 
  subgroup 
  for 
  the 
  I 
  of 
  R. 
  

  

  Since 
  a 
  transformer 
  must 
  permute 
  at 
  least 
  half 
  of 
  the 
  sub- 
  

   stitutions 
  of 
  a 
  group 
  if 
  it 
  permutes 
  one, 
  it 
  follows 
  directly 
  

   from 
  the 
  preceding 
  paragraph 
  that 
  the 
  class 
  of 
  the 
  group 
  of 
  

  

  * 
  Holder, 
  Mathemutische 
  Annalen, 
  vol. 
  xlvi. 
  p. 
  326. 
  

   Phil. 
  Mag. 
  S. 
  5. 
  Vol. 
  45. 
  No. 
  274. 
  March 
  1898. 
  S 
  

  

  