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Theorem List for Metamath Proof Explorer - 15401-15500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfrgpup3lem 15401* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Word        ~FG        freeGrp              g        varFGrp

Theoremfrgpup3 15402* Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
freeGrp              varFGrp

Theorem0frgp 15403 The free group on zero generators is trivial. (Contributed by Mario Carneiro, 21-Apr-2016.)
freeGrp

10.3  Abelian groups

10.3.1  Definition and basic properties

Syntaxccmn 15404 Extend class notation with class of all commutative monoids.
CMnd

Syntaxcabel 15405 Extend class notation with class of all Abelian groups.

Definitiondf-cmn 15406* Define class of all commutative monoids. (Contributed by Mario Carneiro, 6-Jan-2015.)
CMnd

Definitiondf-abl 15407 Define class of all Abelian groups. (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
CMnd

Theoremisabl 15408 The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.)
CMnd

Theoremablgrp 15409 An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)

Theoremablcmn 15410 An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
CMnd

Theoremiscmn 15411* The predicate "is a commutative monoid." (Contributed by Mario Carneiro, 6-Jan-2015.)
CMnd

Theoremisabl2 15412* The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremcmnpropd 15413* If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)
CMnd CMnd

Theoremablpropd 15414* If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 6-Dec-2014.)

Theoremablprop 15415 If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 11-Oct-2013.)

Theoremiscmnd 15416* Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
CMnd

Theoremisabld 15417* Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.)

Theoremisabli 15418* Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.)

Theoremcmnmnd 15419 A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
CMnd

Theoremcmncom 15420 A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.)
CMnd

Theoremablcom 15421 An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.)

Theoremcmn32 15422 Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
CMnd

Theoremcmn4 15423 Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
CMnd

Theoremcmn12 15424 Commutative/associative law for Abelian monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
CMnd

Theoremabl32 15425 Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)

Theoremablinvadd 15426 The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.)

Theoremablsub2inv 15427 Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.)

Theoremablsubadd 15428 Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.)

Theoremablsub4 15429 Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.)

Theoremabladdsub 15431 Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.)

Theoremablpncan2 15432 Cancellation law for subtraction. (Contributed by NM, 2-Oct-2014.)

Theoremablpncan3 15433 A cancellation law for commutative groups. (Contributed by NM, 23-Mar-2015.)

Theoremablsubsub 15434 Law for double subtraction. (Contributed by NM, 7-Apr-2015.)

Theoremablsubsub4 15435 Law for double subtraction. (Contributed by NM, 7-Apr-2015.)

Theoremablpnpcan 15436 Cancellation law for mixed addition and subtraction. (pnpcan 9332 analog.) (Contributed by NM, 29-May-2015.)

Theoremablnncan 15437 Cancellation law for group division. (nncan 9322 analog.) (Contributed by NM, 7-Apr-2015.)

Theoremablsub32 15438 Swap the second and third terms in a double subtraction. (Contributed by NM, 7-Apr-2015.)

Theoremablnnncan1 15439 Cancellation law for subtraction. (nnncan1 9329 analog.) (Contributed by NM, 7-Apr-2015.)

Theoremmulgnn0di 15440 Group multiple of a sum, for nonnegative multiples. (Contributed by Mario Carneiro, 13-Dec-2014.)
.g              CMnd

Theoremmulgdi 15441 Group multiple of a sum. (Contributed by Mario Carneiro, 13-Dec-2014.)
.g

Theoremmulgmhm 15442* The map from to for a fixed positive integer is a monoid homomorphism if the monoid is commutative. (Contributed by Mario Carneiro, 4-May-2015.)
.g       CMnd MndHom

Theoremmulgghm 15443* The map from to for a fixed integer is a group homomorphism if the group is commutative. (Contributed by Mario Carneiro, 4-May-2015.)
.g

Theoremmulgsubdi 15444 Group multiple of a difference. (Contributed by Mario Carneiro, 13-Dec-2014.)
.g

Theoreminvghm 15445 The inversion map is a group automorphism if and only if the group is abelian. (In general it is only a group homomorphism into the opposite group, but in an abelian group the opposite group coincides with the group itself.) (Contributed by Mario Carneiro, 4-May-2015.)

Theoremeqgabl 15446 Value of the subgroup coset equivalence relation on an abelian group. (Contributed by Mario Carneiro, 14-Jun-2015.)
~QG

Theoremsubgabl 15447 A subgroup of an abelian group is also abelian. (Contributed by Mario Carneiro, 3-Dec-2014.)
s        SubGrp

Theoremsubcmn 15448 A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.)
s        CMnd CMnd

Theoremsubmcmn 15449 A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 24-Apr-2016.)
s        CMnd SubMnd CMnd

Theoremsubmcmn2 15450 A submonoid is commutative iff it is a subset of its own centralizer. (Contributed by Mario Carneiro, 24-Apr-2016.)
s        Cntz       SubMnd CMnd

Theoremcntzcmn 15451 The centralizer of any subset in a commutative monoid is the whole monoid. (Contributed by Mario Carneiro, 3-Oct-2015.)
Cntz       CMnd

Theoremcntzspan 15452 If the generators commute, the generated monoid is commutative. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz       mrClsSubMnd       s        CMnd

Theoremghmplusg 15453 The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremablnsg 15454 Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
NrmSGrp SubGrp

Theoremodadd1 15455 The order of a product in an abelian group divides the LCM of the orders of the factors. (Contributed by Mario Carneiro, 20-Oct-2015.)

Theoremodadd2 15456 The order of a product in an abelian group is divisible by the LCM of the orders of the factors divided by the GCD. (Contributed by Mario Carneiro, 20-Oct-2015.)

Theoremodadd 15457 The order of a product is the product of the orders, if the factors have coprime order. (Contributed by Mario Carneiro, 20-Oct-2015.)

Theoremgex2abl 15458 A group with exponent 2 (or 1) is abelian. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx

Theoremgexexlem 15459* Lemma for gexex 15460. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx

Theoremgexex 15460* In an abelian group with finite exponent, there is an element in the group with order equal to the exponent. In other words, all orders of elements divide the largest order of an element of the group. This fails if , for example in an infinite p-group, where there are elements of arbitrarily large orders (so is zero) but no elements of infinite order. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx

Theoremtorsubg 15461 The set of all elements of finite order forms a subgroup of any abelian group, called the torsion subgroup. (Contributed by Mario Carneiro, 20-Oct-2015.)
SubGrp

Theoremoddvdssubg 15462* The set of all elements whose order divides a fixed integer is a subgroup of any abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.)
SubGrp

Theoremlsmcomx 15463 Subgroup sum commutes (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)

Theoremablcntzd 15464 All subgroups in an abelian group commute. (Contributed by Mario Carneiro, 19-Apr-2016.)
Cntz              SubGrp       SubGrp

Theoremlsmcom 15465 Subgroup sum commutes. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
SubGrp SubGrp

Theoremlsmsubg2 15466 The sum of two subgroups is a subgroup. (Contributed by NM, 4-Feb-2014.) (Proof shortened by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp SubGrp

Theoremlsm4 15467 Commutative/associative law for subgroup sum. (Contributed by NM, 26-Sep-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp SubGrp SubGrp

Theoremprdscmnd 15468 The product of a family of commutative monoids is commutative. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s                     CMnd       CMnd

Theoremprdsabld 15469 The product of a family of Abelian groups is an Abelian group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theorempwscmn 15470 The structure power on a commutative monoid is commutative. (Contributed by Mario Carneiro, 11-Jan-2015.)
s        CMnd CMnd

Theorempwsabl 15471 The structure power on an Abelian group is Abelian. (Contributed by Mario Carneiro, 21-Jan-2015.)
s

Theoremdivsabl 15472 If is a subgroup of the abelian group , then is an abelian group. (Contributed by Mario Carneiro, 26-Apr-2016.)
s ~QG        SubGrp

Theoremcnaddablx 15473 The complex numbers are an Abelian group under addition. This version of cnaddabl 15474 shows the explicit structure "scaffold" we chose for the definition for Abelian groups. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use cnaddabl 15474 instead. (New usage is discouraged.) (Contributed by NM, 18-Oct-2012.)

Theoremcnaddabl 15474 The complex numbers are an Abelian group under addition. This version of cnaddablx 15473 hides the explicit structure indices i.e. is "scaffold-independent". Note that the proof also does not reference explicit structure indices. The actual structure is dependent on how and is defined. This theorem should not be referenced in any proof. For the group/ring properties of the complex numbers, see cnrng 16715. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.)

Theoremzaddablx 15475 The integers are an Abelian group under addition. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. Use zsubrg 16744 instead. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)

Theoremfrgpnabllem1 15476* Lemma for frgpnabl 15478. (Contributed by Mario Carneiro, 21-Apr-2016.)
freeGrp       Word        ~FG                      splice               varFGrp

Theoremfrgpnabllem2 15477* Lemma for frgpnabl 15478. (Contributed by Mario Carneiro, 21-Apr-2016.)
freeGrp       Word        ~FG                      splice               varFGrp

Theoremfrgpnabl 15478 The free group on two or more generators is not abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
freeGrp

10.3.2  Cyclic groups

Syntaxccyg 15479 Cyclic group.
CycGrp

Definitiondf-cyg 15480* Define a cyclic group, which is a group with an element , called the generator of the group, such that all elements in the group are multiples of . A generator is usually not unique. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp .g

Theoremiscyg 15481* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g       CycGrp

Theoremiscyggen 15482* The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g

Theoremiscyggen2 15483* The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g

Theoremiscyg2 15484* A cyclic group is a group which contains a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g              CycGrp

Theoremcyggeninv 15485* The inverse of a cyclic generator is a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g

Theoremcyggenod 15486* An element is the generator of a finite group iff the order of the generator equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g

Theoremcyggenod2 15487* In an infinite cyclic group, the generator must have infinite order, but this property no longer characterizes the generators. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g

Theoremiscyg3 15488* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g       CycGrp

Theoremiscygd 15489* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g                            CycGrp

Theoremiscygodd 15490 Show that a group with an element the same order as the group is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
CycGrp

Theoremcyggrp 15491 A cyclic group is a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp

Theoremcygabl 15492 A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp

Theoremcygctb 15493 A cyclic group is countable. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp

Theorem0cyg 15494 The trivial group is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp

Theoremprmcyg 15495 A group with prime order is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
CycGrp

Theoremlt6abl 15496 A group with fewer than elements is abelian. (Contributed by Mario Carneiro, 27-Apr-2016.)

Theoremghmcyg 15497 The image of a cyclic group under a surjective group homomorphism is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp CycGrp

Theoremcyggex2 15498 The exponent of a cyclic group is if the group is infinite, otherwise it equals the order of the group. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx       CycGrp

Theoremcyggex 15499 The exponent of a finite cyclic group is the order of the group. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx       CycGrp

Theoremcyggexb 15500 A finite abelian group is cyclic iff the exponent equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.)
gEx       CycGrp

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