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Theorem List for Metamath Proof Explorer - 15101-15200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremoppglem 15101 Lemma for oppgbas 15102. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg       Slot

Theoremoppgbas 15102 Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg

Theoremoppgtset 15103 Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
oppg       TopSet       TopSet

Theoremoppgtopn 15104 Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
oppg

Theoremoppgmnd 15105 The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
oppg

Theoremoppgmndb 15106 Bidirectional form of oppgmnd 15105. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg

Theoremoppgid 15107 Zero in a monoid is a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
oppg

Theoremoppggrp 15108 The opposite of a group is a group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg

Theoremoppggrpb 15109 Bidirectional form of oppggrp 15108. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg

Theoremoppginv 15110 Inverses in a group are a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg

Theoreminvoppggim 15111 The inverse is an antiautomorphism on any group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg              GrpIso

Theoremoppggic 15112 Every group is (naturally) isomorphic to its opposite. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg       𝑔

Theoremoppgsubm 15113 Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
oppg       SubMnd SubMnd

Theoremoppgsubg 15114 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
oppg       SubGrp SubGrp

Theoremoppgcntz 15115 A centralizer in a group is the same as the centralizer in the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
oppg       Cntz       Cntz

Theoremoppgcntr 15116 The center of a group is the same as the center of the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
oppg       Cntr       Cntr

Theoremgsumwrev 15117 A sum in an opposite monoid is the regular sum of a reversed word. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Proof shortened by Mario Carneiro, 28-Feb-2016.)
oppg       Word g g reverse

10.2.9  p-Groups and Sylow groups; Sylow's theorems

Syntaxcod 15118 Extend class notation to include the order function on the elements of a group.

Syntaxcgex 15119 Extend class notation to include the order function on the elements of a group.
gEx

Syntaxcpgp 15120 Extend class notation to include the class of all p-groups.
pGrp

Syntaxcslw 15121 Extend class notation to include the class of all Sylow p-subgroups of a group.
pSyl

Definitiondf-od 15122* Define the order of an element in a group. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 4-Sep-2015.)
.g

Definitiondf-gex 15123* Define the exponent of a group. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 4-Sep-2015.)
gEx .g

Definitiondf-pgp 15124* Define the set of p-groups, which are groups such that every element has a power of as its order. (Contributed by Mario Carneiro, 15-Jan-2015.)
pGrp

Definitiondf-slw 15125* Define the set of Sylow p-subgroups of a group . A Sylow p-subgroup is a p-group that is not a subgroup of any other p-groups in . (Contributed by Mario Carneiro, 16-Jan-2015.)
pSyl SubGrp SubGrp pGrp s

Theoremodfval 15126* Value of the order function. (Contributed by Mario Carneiro, 13-Jul-2014.)
.g

Theoremodval 15127* Second substitution for the group order definition. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 5-Sep-2015.)
.g

Theoremodlem1 15128* The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
.g

Theoremodcl 15129 The order of a group element is always a nonnegative integer. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)

Theoremodf 15130 Functionality of the group element order. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremodid 15131 Any element to the power of its order is the identity. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
.g

Theoremodlem2 15132 Any positive annihilator of a group element is an upper bound on the (positive) order of the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
.g

Theoremodmodnn0 15133 Reduce the argument of a group multiple by modding out the order of the element. (Contributed by Mario Carneiro, 23-Sep-2015.)
.g

Theoremmndodconglem 15134 Lemma for mndodcong 15135. (Contributed by Mario Carneiro, 23-Sep-2015.)
.g

Theoremmndodcong 15135 If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. (Contributed by Mario Carneiro, 23-Sep-2015.)
.g

Theoremmndodcongi 15136 If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. For monoids, the reverse implication is false for elements with infinite order. For example, the powers of mod are 1,2,4,8,6,2,4,8,6,... so that the identity 1 never repeats, which is infinite order by our definition, yet other numbers like 6 appear many times in the sequence. (Contributed by Mario Carneiro, 23-Sep-2015.)
.g

Theoremoddvdsnn0 15137 The only multiples of that are equal to the identity are the multiples of the order of . (Contributed by Mario Carneiro, 23-Sep-2015.)
.g

Theoremodnncl 15138 If a nonzero multiple of an element is zero, the element has positive order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
.g

Theoremodmod 15139 Reduce the argument of a group multiple by modding out the order of the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 6-Sep-2015.)
.g

Theoremoddvds 15140 The only multiples of that are equal to the identity are the multiples of the order of . (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
.g

Theoremoddvdsi 15141 Any group element is annihilated by any multiple of its order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
.g

Theoremodcong 15142 If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. (Contributed by Stefan O'Rear, 5-Sep-2015.)
.g

Theoremodeq 15143* The oddvds 15140 property uniquely defines the group order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
.g

Theoremodval2 15144* A non-conditional definition of the group order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
.g

Theoremodmulgid 15145 A relationship between the order of a multiple and the order of the basepoint. (Contributed by Stefan O'Rear, 6-Sep-2015.)
.g

Theoremodmulg2 15146 The order of a multiple divides the order of the base point. (Contributed by Stefan O'Rear, 6-Sep-2015.)
.g

Theoremodmulg 15147 Relationship between the order of an element and that of a multiple. (Contributed by Stefan O'Rear, 6-Sep-2015.)
.g

Theoremodmulgeq 15148 A multiple of a point of finite order only has the same order if the multiplier is relatively prime. (Contributed by Stefan O'Rear, 12-Sep-2015.)
.g

Theoremodbezout 15149* If is coprime to the order of , there is a modular inverse to cancel multiplication by . (Contributed by Mario Carneiro, 27-Apr-2016.)
.g

Theoremod1 15150 The order of the group identity is one. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)

Theoremodeq1 15151 The group identity is the unique element of a group with order one. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)

Theoremodinv 15152 The order of the inverse of a group element. (Contributed by Mario Carneiro, 20-Oct-2015.)

Theoremodf1 15153* The multiples of an element with infinite order form an infinite cyclic subgroup of . (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
.g

Theoremodinf 15154* The multiples of an element with infinite order form an infinite cyclic subgroup of . (Contributed by Mario Carneiro, 14-Jan-2015.)
.g

Theoremdfod2 15155* An alternative definition of the order of a group element is as the cardinality of the cyclic subgroup generated by the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
.g

Theoremodcl2 15156 The order of an element of a finite group is finite. (Contributed by Mario Carneiro, 14-Jan-2015.)

Theoremoddvds2 15157 The order of an element of a finite group divides the order (cardinality) of the group. Corollary of Lagrange's theorem for the order of a subgroup. (Contributed by Mario Carneiro, 14-Jan-2015.)

Theoremsubmod 15158 The order of an element is the same in a subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
s                      SubMnd

Theoremsubgod 15159 The order of an element is the same in a subgroup. (Contributed by Mario Carneiro, 14-Jan-2015.) (Proof shortened by Stefan O'Rear, 12-Sep-2015.)
s                      SubGrp

Theoremodsubdvds 15160 The order of an element of a subgroup divides the order of the subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
SubGrp

Theoremodf1o1 15161* An element with zero order has infinitely many multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.)
.g              mrClsSubGrp

Theoremodf1o2 15162* An element with nonzero order has as many multiples as its order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
.g              mrClsSubGrp       ..^ ..^

Theoremodhash 15163 An element of zero order generates an infinite subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
mrClsSubGrp

Theoremodhash2 15164 If an element has nonzero order, it generates a subgroup with size equal to the order. (Contributed by Stefan O'Rear, 12-Sep-2015.)
mrClsSubGrp

Theoremodhash3 15165 An element which generates a finite subgroup has order the size of that subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
mrClsSubGrp

Theoremodngen 15166* A cyclic subgroup of size has generators. (Contributed by Stefan O'Rear, 12-Sep-2015.)
mrClsSubGrp

Theoremgexval 15167* Value of the exponent of a group. (Contributed by Mario Carneiro, 23-Apr-2016.)
.g              gEx

Theoremgexlem1 15168* The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 23-Apr-2016.)
.g              gEx

Theoremgexcl 15169 The exponent of a group is a nonnegative integer. (Contributed by Mario Carneiro, 23-Apr-2016.)
gEx

Theoremgexid 15170 Any element to the power of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx       .g

Theoremgexlem2 15171* Any positive annihilator of all the group elements is an upper bound on the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx       .g

Theoremgexdvdsi 15172 Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx       .g

Theoremgexdvds 15173* The only that annihilate all the elements of the group are the multiples of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx       .g

Theoremgexdvds2 15174* An integer divides the group exponent iff it divides all the group orders. In other words, the group exponent is the LCM of the orders of all the elements. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx

Theoremgexod 15175 Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx

Theoremgexcl3 15176* If the order of every group element is bounded by , the group has finite exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx

Theoremgexnnod 15177 Every group element has finite order if the exponent is finite. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx

Theoremgexcl2 15178 The exponent of a finite group is finite. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx

Theoremgexdvds3 15179 The exponent of a finite group divides the order (cardinality) of the group. Corollary of Lagrange's theorem for the order of a subgroup. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx

Theoremgex1 15180 A group or monoid has exponent 1 iff it is trivial. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx

Theoremispgp 15181* A group is a -group if every element has some power of as its order. (Contributed by Mario Carneiro, 15-Jan-2015.)
pGrp

Theorempgpprm 15182 Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
pGrp

Theorempgpgrp 15183 Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
pGrp

Theorempgpfi1 15184 A finite group with order a power of a prime is a -group. (Contributed by Mario Carneiro, 16-Jan-2015.)
pGrp

Theorempgp0 15185 The identity subgroup is a -group for every prime . (Contributed by Mario Carneiro, 19-Jan-2015.)
pGrp s

Theoremsubgpgp 15186 A subgroup of a p-group is a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
pGrp SubGrp pGrp s

Theoremsylow1lem1 15187* Lemma for sylow1 15192. The p-adic valuation of the size of is equal to the number of excess powers of in . (Contributed by Mario Carneiro, 15-Jan-2015.)

Theoremsylow1lem2 15188* Lemma for sylow1 15192. The function is a group action on . (Contributed by Mario Carneiro, 15-Jan-2015.)

Theoremsylow1lem3 15189* Lemma for sylow1 15192. One of the orbits of the group action has p-adic valuation less than the prime count of the set . (Contributed by Mario Carneiro, 15-Jan-2015.)

Theoremsylow1lem4 15190* Lemma for sylow1 15192. The stabilizer subgroup of any element of is at most in size. (Contributed by Mario Carneiro, 15-Jan-2015.)

Theoremsylow1lem5 15191* Lemma for sylow1 15192. Using Lagrange's theorem and the orbit-stabilizer theorem, show that there is a subgroup with size exactly . (Contributed by Mario Carneiro, 16-Jan-2015.)
SubGrp

Theoremsylow1 15192* Sylow's first theorem. If is a prime power that divides the cardinality of , then has a supgroup with size . (Contributed by Mario Carneiro, 16-Jan-2015.)
SubGrp

Theoremodcau 15193* Cauchy's theorem for the order of an element in a group. A finite group whose order divides a prime contains an element of order . (Contributed by Mario Carneiro, 16-Jan-2015.)

Theorempgpfi 15194* The converse to pgpfi1 15184. A finite group is a -group iff it has size some power of . (Contributed by Mario Carneiro, 16-Jan-2015.)
pGrp

Theorempgpfi2 15195 Alternate version of pgpfi 15194. (Contributed by Mario Carneiro, 27-Apr-2016.)
pGrp

Theorempgphash 15196 The order of a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
pGrp

Theoremisslw 15197* The property of being a Sylow subgroup. A Sylow -subgroup is a -group which has no proper supersets that are also -groups. (Contributed by Mario Carneiro, 16-Jan-2015.)
pSyl SubGrp SubGrp pGrp s

Theoremslwprm 15198 Reverse closure for the first argument of a Sylow -subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 2-May-2015.)
pSyl

Theoremslwsubg 15199 A Sylow -subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
pSyl SubGrp

Theoremslwispgp 15200 Defining property of a Sylow -subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
s        pSyl SubGrp pGrp

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