:: Cyclic Groups and Some of Their Properties - Part I
:: by Dariusz Surowik
::
:: Received November 22, 1991
:: Copyright (c) 1991-2021 Association of Mizar Users

Lm1: for x being set
for n being Nat st x in Segm n holds
x is Element of NAT
;

definition

redefine func addint means :: GR_CY_1:def 1
for i1, i2 being Element of INT holds it . (i1,i2) = addreal . (i1,i2);

compatibility
for b₁ being Element of bool [:[:INT,INT:],INT:] holds
( b₁ = addint iff for i1, i2 being Element of INT holds b₁ . (i1,i2) = addreal . (i1,i2) )

proof end;

end;

:: deftheorem defines addint GR_CY_1:def 1 :
for b₁ being Element of bool [:[:INT,INT:],INT:] holds
( b₁ = addint iff for i1, i2 being Element of INT holds b₁ . (i1,i2) = addreal . (i1,i2) );

theorem :: GR_CY_1:1

for i1 being Element of INT st i1 = 0 holds
i1 is_a_unity_wrt addint by BINOP_2:4, SETWISEO:14;

theorem Th2: :: GR_CY_1:2

for I being FinSequence of INT holds Sum I = addint $$ I

proof end;

definition

let I be FinSequence of INT ;

:: original: Sum
redefine func Sum I -> Element of INT equals :: GR_CY_1:def 2
addint $$ I;

coherence
Sum I is Element of INT

proof end;

compatibility
for b₁ being Element of INT holds
( b₁ = Sum I iff b₁ = addint $$ I ) by Th2;

end;

:: deftheorem defines Sum GR_CY_1:def 2 :
for I being FinSequence of INT holds Sum I = addint $$ I;

theorem Th3: :: GR_CY_1:3

Sum (<*> INT) = 0 by BINOP_2:4, FINSOP_1:10;

Lm2: for G being Group
for a being Element of G holds Product (((len (<*> INT)) |-> a) |^ (<*> INT)) = a |^ (Sum (<*> INT))

proof end;

Lm3: for G being Group
for a being Element of G
for I being FinSequence of INT
for w being Element of INT st Product (((len I) |-> a) |^ I) = a |^ (Sum I) holds
Product (((len (I ^ <*w*>)) |-> a) |^ (I ^ <*w*>)) = a |^ (Sum (I ^ <*w*>))

proof end;

theorem Th4: :: GR_CY_1:4

for G being Group
for a being Element of G
for I being FinSequence of INT holds Product (((len I) |-> a) |^ I) = a |^ (Sum I)

proof end;

:: Finite groups and their some properties

theorem Th5: :: GR_CY_1:5

for G being Group
for a, b being Element of G holds
( b in gr {a} iff ex j1 being Integer st b = a |^ j1 )

proof end;

theorem Th6: :: GR_CY_1:6

for G being finite Group
for a being Element of G holds not a is being_of_order_0

proof end;

theorem Th7: :: GR_CY_1:7

for G being finite Group
for a being Element of G holds ord a = card (gr {a})

proof end;

theorem Th8: :: GR_CY_1:8

for G being finite Group
for a being Element of G holds ord a divides card G

proof end;

theorem Th9: :: GR_CY_1:9

for G being finite Group
for a being Element of G holds a |^ (card G) = 1_ G

proof end;

theorem Th10: :: GR_CY_1:10

for n being Nat
for G being finite Group
for a being Element of G holds (a |^ n) " = a |^ ((card G) - (n mod (card G)))

proof end;

registration

let G be non empty associative multMagma ;

cluster multMagma(# the carrier of G, the multF of G #) -> associative ;

coherence
multMagma(# the carrier of G, the multF of G #) is associative by BINOP_1:def 3;

end;

registration

let G be Group;

cluster multMagma(# the carrier of G, the multF of G #) -> Group-like ;

coherence
multMagma(# the carrier of G, the multF of G #) is Group-like

proof end;

end;

theorem Th11: :: GR_CY_1:11

for G being finite strict Group st card G > 1 holds
ex a being Element of G st a <> 1_ G

proof end;

theorem :: GR_CY_1:12

for p being Nat
for G being finite strict Group st card G = p & p is prime holds
for H being strict Subgroup of G holds
( H = (1). G or H = G )

proof end;

definition

func INT.Group -> non empty strict multMagma equals :: GR_CY_1:def 3
multMagma(# INT,addint #);

coherence
multMagma(# INT,addint #) is non empty strict multMagma ;

end;

:: deftheorem defines INT.Group GR_CY_1:def 3 :
INT.Group = multMagma(# INT,addint #);

registration

cluster INT.Group -> non empty strict Group-like associative ;

coherence
( INT.Group is associative & INT.Group is Group-like )

proof end;

end;

registration

cluster the carrier of INT.Group -> integer-membered ;

coherence
the carrier of INT.Group is integer-membered ;

end;

registration

let a, b be Element of INT.Group;

identify a * b with a + b;

compatibility
a * b = a + b by BINOP_2:def 20;

end;

registration

let n be natural Number ;

cluster Segm n -> natural-membered ;

coherence
Segm n is natural-membered ;

end;

definition

let n be natural Number ;
assume A1: n > 0 ;

func addint n -> BinOp of (Segm n) means :Def4: :: GR_CY_1:def 4
for k, l being Element of Segm n holds it . (k,l) = (k + l) mod n;

existence
ex b₁ being BinOp of (Segm n) st
for k, l being Element of Segm n holds b₁ . (k,l) = (k + l) mod n

proof end;

uniqueness
for b₁, b₂ being BinOp of (Segm n) st ( for k, l being Element of Segm n holds b₁ . (k,l) = (k + l) mod n ) & ( for k, l being Element of Segm n holds b₂ . (k,l) = (k + l) mod n ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines addint GR_CY_1:def 4 :
for n being natural Number st n > 0 holds
for b₂ being BinOp of (Segm n) holds
( b₂ = addint n iff for k, l being Element of Segm n holds b₂ . (k,l) = (k + l) mod n );

definition

let n be non zero Nat;

func INT.Group n -> non empty strict multMagma equals :: GR_CY_1:def 5
multMagma(# (Segm n),(addint n) #);

coherence
multMagma(# (Segm n),(addint n) #) is non empty strict multMagma ;

end;

:: deftheorem defines INT.Group GR_CY_1:def 5 :
for n being non zero Nat holds INT.Group n = multMagma(# (Segm n),(addint n) #);

registration

let n be non zero Nat;

cluster INT.Group n -> non empty finite strict Group-like associative ;

coherence
( INT.Group n is finite & INT.Group n is associative & INT.Group n is Group-like )

proof end;

end;

theorem Th13: :: GR_CY_1:13

1_ INT.Group = 0

proof end;

theorem Th14: :: GR_CY_1:14

for n being non zero Nat holds 1_ (INT.Group n) = 0

proof end;

definition

let h be Integer;

func @' h -> Element of INT.Group equals :: GR_CY_1:def 6
h;

coherence
h is Element of INT.Group by INT_1:def 2;

end;

:: deftheorem defines @' GR_CY_1:def 6 :
for h being Integer holds @' h = h;

theorem Th15: :: GR_CY_1:15

for h being Element of INT.Group holds h " = - h

proof end;

Lm4: for k being Nat holds (@' 1) |^ k = k

proof end;

theorem Th16: :: GR_CY_1:16

for j1 being Integer holds j1 = (@' 1) |^ j1

proof end;

Lm5: INT.Group = gr {(@' 1)}

proof end;

definition

let IT be Group;

attr IT is cyclic means :Def7: :: GR_CY_1:def 7
ex a being Element of IT st multMagma(# the carrier of IT, the multF of IT #) = gr {a};

end;

:: deftheorem Def7 defines cyclic GR_CY_1:def 7 :
for IT being Group holds
( IT is cyclic iff ex a being Element of IT st multMagma(# the carrier of IT, the multF of IT #) = gr {a} );

registration

cluster non empty strict unital Group-like associative cyclic for multMagma ;

existence
ex b₁ being Group st
( b₁ is strict & b₁ is cyclic ) by Def7, Lm5;

end;

registration

let G be Group;

cluster (1). G -> cyclic ;

coherence
(1). G is cyclic

proof end;

end;

registration

cluster non empty finite strict unital Group-like associative cyclic for multMagma ;

existence
ex b₁ being Group st
( b₁ is strict & b₁ is finite & b₁ is cyclic )

proof end;

end;

theorem Th17: :: GR_CY_1:17

for G being Group holds
( G is cyclic Group iff ex a being Element of G st
for b being Element of G ex j1 being Integer st b = a |^ j1 )

proof end;

theorem Th18: :: GR_CY_1:18

for G being finite Group holds
( G is cyclic iff ex a being Element of G st
for b being Element of G ex n being Nat st b = a |^ n )

proof end;

theorem Th19: :: GR_CY_1:19

for G being finite Group holds
( G is cyclic iff ex a being Element of G st ord a = card G )

proof end;

theorem :: GR_CY_1:20

for G being finite Group
for H being strict Subgroup of G st G is cyclic holds
H is cyclic

proof end;

theorem :: GR_CY_1:21

for G being finite strict Group st card G is prime holds
G is cyclic

proof end;

theorem Th22: :: GR_CY_1:22

for n being non zero Nat ex g being Element of (INT.Group n) st
for b being Element of (INT.Group n) ex j1 being Integer st b = g |^ j1

proof end;

registration

cluster non empty Group-like associative cyclic -> commutative for multMagma ;

coherence
for b₁ being Group st b₁ is cyclic holds
b₁ is commutative

proof end;

end;

theorem :: GR_CY_1:23

INT.Group = gr {(@' 1)} by Lm5;

registration

cluster INT.Group -> non empty strict cyclic ;

coherence
INT.Group is cyclic by Lm5;

end;

registration

let n be non zero Nat;

cluster INT.Group n -> non empty strict cyclic ;

coherence
INT.Group n is cyclic

proof end;

end;