Volume 3, 1991

University of Bialystok

Copyright (c) 1991 Association of Mizar Users

### The abstract of the Mizar article:

### Cyclic Groups and Some of Their Properties --- Part I

**by****Dariusz Surowik**- Received November 22, 1991
- MML identifier: GR_CY_1

- [ Mizar article, MML identifier index ]

environ vocabulary INT_1, REALSET1, FINSEQ_1, NAT_1, ORDINAL2, ARYTM, BINOP_1, FUNCT_1, VECTSP_1, QC_LANG1, SETWISEO, RLVECT_1, FINSEQ_2, GROUP_1, GROUP_4, RELAT_1, ARYTM_1, FUNCOP_1, FINSET_1, CARD_1, GROUP_2, TARSKI, ARYTM_3, FILTER_0, RLSUB_1, GRAPH_1, GR_CY_1, CARD_3; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL2, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, RELAT_1, FUNCT_1, REAL_1, FUNCT_2, SETWISEO, CARD_1, FINSET_1, BINOP_1, DOMAIN_1, INT_1, INT_2, NAT_1, RLVECT_1, GROUP_2, STRUCT_0, VECTSP_1, GROUP_1, GROUP_4, FUNCOP_1, FINSOP_1, SETWOP_2, FINSEQ_1; constructors WELLORD2, REAL_1, SETWISEO, DOMAIN_1, NAT_1, FUNCSDOM, GROUP_4, FINSOP_1, NAT_LAT, MEMBERED, XBOOLE_0; clusters SUBSET_1, FUNCT_1, CARD_1, INT_1, GROUP_1, GROUP_2, RELSET_1, STRUCT_0, FINSEQ_1, ARYTM_3, XREAL_0, MEMBERED, ZFMISC_1, XBOOLE_0, ORDINAL2; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; begin reserve i1,i2,i3 for Element of INT; reserve j1,j2,j3 for Integer; reserve p,s,r,g,k,n,m,q,t for Nat; reserve x,y,xp,yp for set; reserve G for Group; reserve a,b for Element of G; reserve F for FinSequence of the carrier of G; reserve I for FinSequence of INT; ::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: Some properties of modulo function :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::: theorem :: GR_CY_1:1 m mod n = (n*k + m) mod n; theorem :: GR_CY_1:2 (p+s) mod n = ((p mod n)+s) mod n; theorem :: GR_CY_1:3 (p+s) mod n = (p + ( s mod n)) mod n; theorem :: GR_CY_1:4 k < n implies k mod n = k; theorem :: GR_CY_1:5 n mod n = 0; theorem :: GR_CY_1:6 0 = 0 mod n; definition let n be natural number such that n > 0; func Segm (n) -> non empty Subset of NAT equals :: GR_CY_1:def 1 {p: p<n}; end; canceled 3; theorem :: GR_CY_1:10 for n,s being natural number st n > 0 holds s in Segm (n) iff s < n; canceled; theorem :: GR_CY_1:12 for n being natural number st n > 0 holds 0 in Segm (n); theorem :: GR_CY_1:13 Segm (1) = {0}; ::::::::::::::::::::::::::::::::::::::::::::::::::::: :: Definition addint :: ::::::::::::::::::::::::::::::::::::::::::::::::::::: definition func addint -> BinOp of INT means :: GR_CY_1:def 2 for i1,i2 being Element of INT holds it.(i1,i2) = addreal.(i1,i2); end; theorem :: GR_CY_1:14 for j1,j2 holds addint.(j1,j2)=j1+j2; theorem :: GR_CY_1:15 for i1 st i1 = 0 holds i1 is_a_unity_wrt addint; theorem :: GR_CY_1:16 the_unity_wrt addint = 0; theorem :: GR_CY_1:17 addint has_a_unity; theorem :: GR_CY_1:18 addint is commutative; theorem :: GR_CY_1:19 addint is associative; definition let F be FinSequence of INT; func Sum(F) -> Integer equals :: GR_CY_1:def 3 addint $$ F; end; theorem :: GR_CY_1:20 Sum(I^<*i1*>) =Sum I +@i1; theorem :: GR_CY_1:21 Sum <*i1*> = i1; theorem :: GR_CY_1:22 Sum (<*> INT) = 0; canceled; theorem :: GR_CY_1:24 for I being FinSequence of INT holds Product(((len I)|->a)|^I) = a|^Sum I; ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: Finite groups and their some properties :: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: theorem :: GR_CY_1:25 b in gr {a} iff ex j1 st b=a|^j1; theorem :: GR_CY_1:26 G is finite implies a is_not_of_order_0; theorem :: GR_CY_1:27 G is finite implies ord a = ord gr {a}; theorem :: GR_CY_1:28 G is finite implies ord a divides ord G; theorem :: GR_CY_1:29 G is finite implies a|^ord G = 1.G; theorem :: GR_CY_1:30 G is finite implies (a|^n)" = a|^(ord G - (n mod ord G)); theorem :: GR_CY_1:31 for G being strict Group holds ord G > 1 implies ex a being Element of G st a <> 1.G; theorem :: GR_CY_1:32 for G being strict Group holds G is finite & ord G = p & p is prime implies for H being strict Subgroup of G holds H = (1).G or H = G; theorem :: GR_CY_1:33 HGrStr(#INT,addint#) is associative Group-like; definition func INT.Group -> strict Group equals :: GR_CY_1:def 4 HGrStr(#INT,addint#); end; definition let n such that n > 0; func addint(n) -> BinOp of Segm(n) means :: GR_CY_1:def 5 for k,l being Element of Segm(n) holds it.(k,l) = (k+l) mod n; end; theorem :: GR_CY_1:34 for n st n > 0 holds HGrStr(#Segm(n),addint(n)#) is associative Group-like; definition let n such that n > 0; func INT.Group(n) -> strict Group equals :: GR_CY_1:def 6 HGrStr(#Segm(n),addint(n)#); end; theorem :: GR_CY_1:35 1.INT.Group = 0; theorem :: GR_CY_1:36 for n st n>0 holds 1.INT.Group(n) = 0; definition let h be Element of INT.Group; func @'h -> Integer equals :: GR_CY_1:def 7 h; end; definition let h be Integer; func @'h -> Element of INT.Group equals :: GR_CY_1:def 8 h; end; theorem :: GR_CY_1:37 for h being Element of INT.Group holds h" = -@'h; reserve G1 for Subgroup of INT.Group, h for Element of INT.Group; theorem :: GR_CY_1:38 for h st h=1 holds for k holds h|^k = k; theorem :: GR_CY_1:39 for h,j1 st h=1 holds j1 = h |^ j1; definition let IT be Group; attr IT is cyclic means :: GR_CY_1:def 9 ex a being Element of IT st the HGrStr of IT=gr {a}; end; definition cluster strict cyclic Group; end; theorem :: GR_CY_1:40 (1).G is cyclic; theorem :: GR_CY_1:41 G is cyclic Group iff ex a being Element of G st for b being Element of G ex j1 st b=a|^j1; theorem :: GR_CY_1:42 G is finite implies (G is cyclic Group iff ex a being Element of G st for b being Element of G ex n st b=a|^n); theorem :: GR_CY_1:43 for G being strict Group holds G is finite implies ( G is cyclic Group iff ex a being Element of G st ord a =ord G ); theorem :: GR_CY_1:44 for H being strict Subgroup of G holds G is finite & G is cyclic Group implies H is cyclic Group; theorem :: GR_CY_1:45 for G being strict Group holds G is finite & ord (G) = p & p is prime implies G is cyclic Group; theorem :: GR_CY_1:46 for n st n>0 ex g being Element of INT.Group(n) st for b being Element of INT.Group(n) ex j1 st b = g|^j1; definition cluster cyclic -> commutative Group; end; canceled; theorem :: GR_CY_1:48 INT.Group is cyclic; theorem :: GR_CY_1:49 for n st n > 0 holds INT.Group(n) is cyclic Group; theorem :: GR_CY_1:50 INT.Group is commutative Group; theorem :: GR_CY_1:51 for n st n>0 holds INT.Group(n) is commutative Group;

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