Article GR_CY_2, MML version 4.99.1005

:: GR_CY_2:th 6
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1
for b4 being Element of NAT
      st 1 < ord b2 & b2 = b3 |^ b4
   holds b4 <> {};

:: GR_CY_2:th 8
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
   b2 in gr {b2};

:: GR_CY_2:th 9
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
      st b3 = b4
   holds gr {b3} = gr {b4};

:: GR_CY_2:th 10
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
   gr {b2} is non empty Group-like associative cyclic multMagma;

:: GR_CY_2:th 11
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
      for b3 being Element of the carrier of b1 holds
         ex b4 being integer set st
            b3 = b2 |^ b4
   iff
      b1 = gr {b2};

:: GR_CY_2:th 12
theorem
for b1 being non empty finite strict Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
      for b3 being Element of the carrier of b1 holds
         ex b4 being Element of NAT st
            b3 = b2 |^ b4
   iff
      b1 = gr {b2};

:: GR_CY_2:th 13
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of the carrier of b1
   st b1 is finite & b1 = gr {b2}
for b3 being strict Subgroup of b1 holds
   ex b4 being Element of NAT st
      b3 = gr {b2 |^ b4};

:: GR_CY_2:th 14
theorem
for b1, b2, b3 being Element of NAT
for b4 being non empty finite Group-like associative multMagma
for b5 being Element of the carrier of b4
      st b4 = gr {b5} & ord b4 = b1 & b1 = b2 * b3
   holds ord (b5 |^ b2) = b3;

:: GR_CY_2:th 15
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3, b4 being Element of NAT
      st b3 divides b4
   holds b2 |^ b4 in gr {b2 |^ b3};

:: GR_CY_2:th 16
theorem
for b1, b2 being Element of NAT
for b3 being non empty finite Group-like associative multMagma
for b4 being Element of the carrier of b3
      st ord gr {b4 |^ b1} = ord gr {b4 |^ b2} &
         b4 |^ b2 in gr {b4 |^ b1}
   holds gr {b4 |^ b1} = gr {b4 |^ b2};

:: GR_CY_2:th 17
theorem
for b1, b2, b3 being Element of NAT
for b4 being non empty finite Group-like associative multMagma
for b5 being Subgroup of b4
for b6 being Element of the carrier of b4
      st ord b4 = b1 & b4 = gr {b6} & ord b5 = b2 & b5 = gr {b6 |^ b3}
   holds b1 divides b3 * b2;

:: GR_CY_2:th 18
theorem
for b1, b2 being Element of NAT
for b3 being non empty finite strict Group-like associative multMagma
for b4 being Element of the carrier of b3
      st b3 = gr {b4} & ord b3 = b1
   holds    b3 = gr {b4 |^ b2}
   iff
      b2 hcf b1 = 1;

:: GR_CY_2:th 19
theorem
for b1 being non empty Group-like associative cyclic multMagma
for b2 being Subgroup of b1
for b3 being Element of the carrier of b1
      st b1 = gr {b3} & b3 in b2
   holds multMagma(#the carrier of b1,the multF of b1#) = multMagma(#the carrier of b2,the multF of b2#);

:: GR_CY_2:th 20
theorem
for b1 being non empty Group-like associative cyclic multMagma
for b2 being Element of the carrier of b1
      st b1 = gr {b2}
   holds    b1 is finite
   iff
      ex b3, b4 being integer set st
         b3 <> b4 & b2 |^ b3 = b2 |^ b4;

:: GR_CY_2:funcnot 1 => GR_CY_2:func 1
definition
  let a1 be Element of NAT;
  let a2 be Element of the carrier of INT.Group a1;
  assume {} < a1;
  func @ A2 -> Element of NAT equals
    a2;
end;

:: GR_CY_2:def 1
theorem
for b1 being Element of NAT
   st {} < b1
for b2 being Element of the carrier of INT.Group b1 holds
   @ b2 = b2;

:: GR_CY_2:th 21
theorem
for b1 being non empty finite strict Group-like associative cyclic multMagma holds
   INT.Group ord b1,b1 are_isomorphic;

:: GR_CY_2:th 22
theorem
for b1 being non empty strict Group-like associative cyclic multMagma
      st b1 is infinite
   holds INT.Group,b1 are_isomorphic;

:: GR_CY_2:th 23
theorem
for b1, b2 being non empty finite strict Group-like associative cyclic multMagma
      st ord b2 = ord b1
   holds b2,b1 are_isomorphic;

:: GR_CY_2:th 24
theorem
for b1 being Element of NAT
for b2, b3 being non empty finite strict Group-like associative multMagma
      st ord b2 = b1 & ord b3 = b1 & b1 is prime
   holds b2,b3 are_isomorphic;

:: GR_CY_2:th 25
theorem
for b1, b2 being non empty finite strict Group-like associative multMagma
      st ord b1 = 2 & ord b2 = 2
   holds b1,b2 are_isomorphic;

:: GR_CY_2:th 26
theorem
for b1 being non empty finite strict Group-like associative multMagma
   st ord b1 = 2
for b2 being strict Subgroup of b1
      st not b2 = (1). b1
   holds b2 = b1;

:: GR_CY_2:th 27
theorem
for b1 being non empty finite strict Group-like associative multMagma
      st ord b1 = 2
   holds b1 is non empty Group-like associative cyclic multMagma;

:: GR_CY_2:th 28
theorem
for b1 being Element of NAT
for b2 being non empty finite strict Group-like associative multMagma
   st b2 is cyclic & ord b2 = b1
for b3 being Element of NAT
      st b3 divides b1
   holds ex b4 being strict Subgroup of b2 st
      ord b4 = b3 &
       (for b5 being strict Subgroup of b2
             st ord b5 = b3
          holds b5 = b4);

:: GR_CY_2:th 29
theorem
for b1 being non empty Group-like associative cyclic multMagma
for b2 being Element of the carrier of b1
   st b1 = gr {b2}
for b3 being non empty Group-like associative multMagma
for b4 being Function-like quasi_total multiplicative Relation of the carrier of b3,the carrier of b1
      st b2 in Image b4
   holds b4 is being_epimorphism(b3, b1);

:: GR_CY_2:th 30
theorem
for b1 being non empty finite strict Group-like associative cyclic multMagma
      st ex b2 being Element of NAT st
           ord b1 = 2 * b2
   holds ex b2 being Element of the carrier of b1 st
      ord b2 = 2 &
       (for b3 being Element of the carrier of b1
             st ord b3 = 2
          holds b2 = b3);

:: GR_CY_2:funcreg 1
registration
  let a1 be non empty Group-like associative multMagma;
  cluster center a1 -> strict normal;
end;

:: GR_CY_2:th 31
theorem
for b1 being non empty finite strict Group-like associative cyclic multMagma
      st ex b2 being Element of NAT st
           ord b1 = 2 * b2
   holds ex b2 being Subgroup of b1 st
      ord b2 = 2 & b2 is non empty Group-like associative cyclic multMagma;

:: GR_CY_2:th 32
theorem
for b1 being non empty Group-like associative multMagma
for b2 being non empty strict Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b1
      st b2 is cyclic
   holds Image b3 is cyclic;

:: GR_CY_2:th 33
theorem
for b1, b2 being non empty strict Group-like associative multMagma
      st b1,b2 are_isomorphic & b1 is cyclic
   holds b2 is cyclic;