﻿236 
  Dr. 
  G. 
  A. 
  Miller 
  on 
  the 
  Simple 
  Isomorphisms 
  

  

  morphism 
  of 
  G 
  to 
  itself. 
  Hence 
  it 
  cannot 
  exceed 
  the 
  order 
  

   of 
  G 
  diminished 
  by 
  unity. 
  

  

  If 
  an 
  isomorphism 
  of 
  G 
  to 
  itself 
  can 
  be 
  obtained 
  by 
  trans- 
  

   forming 
  it 
  with 
  respect 
  to 
  one 
  of 
  its 
  own 
  substitutions, 
  it 
  is 
  

   said 
  to 
  be 
  cogredient. 
  All 
  the 
  other 
  simple 
  isomorphisms 
  

   of 
  G 
  to 
  itself 
  are 
  said 
  to 
  be 
  contragredient. 
  Two 
  simple 
  iso- 
  

   morphisms 
  of 
  G 
  to 
  itself 
  which 
  cannot 
  be 
  transformed 
  into 
  

   each 
  other 
  may 
  be 
  called 
  distinct. 
  Hence, 
  the 
  number 
  of 
  

   distinct 
  isomorphisms 
  of 
  G 
  is 
  equal 
  to 
  the 
  number 
  of 
  different 
  

   intransitive 
  groups 
  of 
  twice 
  the 
  degree 
  of 
  G 
  that 
  may 
  be 
  

   formed 
  by 
  making 
  G 
  simple 
  isomorphic 
  to 
  itself. 
  

  

  When 
  two 
  isomorphisms 
  are 
  not 
  distinct, 
  they 
  may 
  be 
  said 
  

   to 
  be 
  transform 
  with 
  respect 
  to 
  each 
  other. 
  All 
  the 
  cogredient 
  

   isomorphisms 
  of 
  G 
  are 
  transforms 
  of 
  the 
  identical, 
  but 
  they 
  

   do 
  not 
  necessarily 
  form 
  a 
  complete 
  system 
  of 
  transforms. 
  

   The 
  cogredient 
  isomorphisms 
  correspond 
  to 
  a 
  self-conjugate 
  

   subgroup 
  in 
  the 
  group 
  of 
  isomorphisms. 
  The 
  subgroup 
  of 
  

   this 
  group, 
  which 
  corresponds 
  to 
  the 
  transforms 
  of 
  the 
  

   identical 
  isomorphism, 
  includes 
  this 
  self-conjugate 
  subgroup 
  ; 
  

   but 
  it 
  is 
  not 
  necessarily 
  self- 
  conjugate. 
  From 
  the 
  fact 
  that 
  

   the 
  transforms 
  of 
  the 
  identical 
  isomorphism 
  correspond 
  to 
  a 
  

   subgroup 
  of 
  the 
  group 
  of 
  isomorphisms, 
  it 
  follows 
  that 
  the 
  

   total 
  number 
  of 
  the 
  simple 
  isomorphisms 
  of 
  G 
  to 
  itself 
  is 
  

   divisible 
  by 
  the 
  number 
  of 
  the 
  transforms 
  of 
  the 
  identical 
  

   isomorphism. 
  

  

  It 
  may 
  happen 
  that 
  a 
  subgroup 
  of 
  G 
  corresponds 
  to 
  itself 
  

   in 
  all 
  the 
  possible 
  simple 
  isomorphisms 
  of 
  G 
  to 
  itself. 
  Fro- 
  

   benius 
  has 
  called 
  such 
  a 
  subgroup 
  characteristic 
  *. 
  Subgroups 
  

   which 
  correspond 
  to 
  each 
  other 
  in 
  any 
  of 
  these 
  isomorphisms 
  

   may 
  be 
  called 
  isomorphic^ 
  and 
  those 
  which 
  correspond 
  in 
  any 
  

   of 
  the 
  transforms 
  of 
  the 
  identical 
  isomorphism 
  may 
  be 
  called 
  

   transform. 
  Hence, 
  the 
  isomorphic 
  subgroups 
  include 
  the 
  

   transform 
  and 
  the 
  latter 
  include 
  the 
  conjugate. 
  

  

  If 
  a 
  subgroup 
  corresponds 
  to 
  itself 
  in 
  all 
  the 
  transforms 
  of 
  

   the 
  identical 
  isomorphism 
  it 
  may 
  be 
  called 
  self 
  -transform. 
  

   Hence 
  the 
  seh-conjugate 
  subgroups 
  include 
  the 
  self-transform, 
  

   and 
  the 
  latter 
  include 
  the 
  characteristic. 
  The 
  last 
  might 
  be 
  

   called 
  self-isomorphic, 
  in 
  harmony 
  with 
  the 
  other 
  notation. 
  

   While 
  no 
  special 
  attention 
  is 
  called 
  to 
  these 
  names, 
  yet 
  it 
  is 
  

   very 
  important 
  to 
  observe 
  the 
  special 
  properties 
  of 
  these 
  six 
  

   types 
  of 
  subgroups. 
  

  

  When 
  a 
  group, 
  like 
  the 
  symmetric 
  groups 
  whose 
  order 
  is 
  

   not 
  720 
  t 
  and 
  all 
  the 
  metacyclic 
  groups, 
  admits 
  only 
  co- 
  

  

  * 
  Sitzuvgsberichte 
  der 
  Berliner 
  Akademie, 
  1895, 
  p. 
  183. 
  

   t 
  Holder, 
  Mathemcdisehe 
  Anualen, 
  vol. 
  xlvi. 
  p. 
  345 
  ; 
  Miller, 
  'Bulletin 
  

   of 
  the 
  American 
  Mathematical 
  Society 
  ' 
  (1895), 
  vol. 
  i. 
  p. 
  258, 
  

  

  