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

Theoremmulgmhm 15401* 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 15402* 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 15403 Group multiple of a difference. (Contributed by Mario Carneiro, 13-Dec-2014.)
.g

Theoreminvghm 15404 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 15405 Value of the subgroup coset equivalence relation on an abelian group. (Contributed by Mario Carneiro, 14-Jun-2015.)
~QG

Theoremsubgabl 15406 A subgroup of an abelian group is also abelian. (Contributed by Mario Carneiro, 3-Dec-2014.)
s        SubGrp

Theoremsubcmn 15407 A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.)
s        CMnd CMnd

Theoremsubmcmn 15408 A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 24-Apr-2016.)
s        CMnd SubMnd CMnd

Theoremsubmcmn2 15409 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 15410 The centralizer of any subset in a commutative monoid is the whole monoid. (Contributed by Mario Carneiro, 3-Oct-2015.)
Cntz       CMnd

Theoremcntzspan 15411 If the generators commute, the generated monoid is commutative. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz       mrClsSubMnd       s        CMnd

Theoremghmplusg 15412 The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremablnsg 15413 Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
NrmSGrp SubGrp

Theoremodadd1 15414 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 15415 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 15416 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 15417 A group with exponent 2 (or 1) is abelian. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx

Theoremgexexlem 15418* Lemma for gexex 15419. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx

Theoremgexex 15419* 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 15420 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 15421* 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 15422 Subgroup sum commutes (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)

Theoremablcntzd 15423 All subgroups in an abelian group commute. (Contributed by Mario Carneiro, 19-Apr-2016.)
Cntz              SubGrp       SubGrp

Theoremlsmcom 15424 Subgroup sum commutes. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
SubGrp SubGrp

Theoremlsmsubg2 15425 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 15426 Commutative/associative law for subgroup sum. (Contributed by NM, 26-Sep-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
SubGrp SubGrp SubGrp SubGrp

Theoremprdscmnd 15427 The product of a family of commutative monoids is commutative. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s                     CMnd       CMnd

Theoremprdsabld 15428 The product of a family of Abelian groups is an Abelian group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theorempwscmn 15429 The structure power on a commutative monoid is commutative. (Contributed by Mario Carneiro, 11-Jan-2015.)
s        CMnd CMnd

Theorempwsabl 15430 The structure power on an Abelian group is Abelian. (Contributed by Mario Carneiro, 21-Jan-2015.)
s

Theoremdivsabl 15431 If is a subgroup of the abelian group , then is an abelian group. (Contributed by Mario Carneiro, 26-Apr-2016.)
s ~QG        SubGrp

Theoremcnaddablx 15432 The complex numbers are an Abelian group under addition. This version of cnaddabl 15433 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 15433 instead. (New usage is discouraged.) (Contributed by NM, 18-Oct-2012.)

Theoremcnaddabl 15433 The complex numbers are an Abelian group under addition. This version of cnaddablx 15432 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 16674. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.)

Theoremzaddablx 15434 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 16703 instead. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)

Theoremfrgpnabllem1 15435* Lemma for frgpnabl 15437. (Contributed by Mario Carneiro, 21-Apr-2016.)
freeGrp       Word        ~FG                      splice               varFGrp

Theoremfrgpnabllem2 15436* Lemma for frgpnabl 15437. (Contributed by Mario Carneiro, 21-Apr-2016.)
freeGrp       Word        ~FG                      splice               varFGrp

Theoremfrgpnabl 15437 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 15438 Cyclic group.
CycGrp

Definitiondf-cyg 15439* 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 15440* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g       CycGrp

Theoremiscyggen 15441* The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g

Theoremiscyggen2 15442* The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g

Theoremiscyg2 15443* A cyclic group is a group which contains a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g              CycGrp

Theoremcyggeninv 15444* The inverse of a cyclic generator is a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g

Theoremcyggenod 15445* 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 15446* 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 15447* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g       CycGrp

Theoremiscygd 15448* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g                            CycGrp

Theoremiscygodd 15449 Show that a group with an element the same order as the group is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
CycGrp

Theoremcyggrp 15450 A cyclic group is a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp

Theoremcygabl 15451 A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp

Theoremcygctb 15452 A cyclic group is countable. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp

Theorem0cyg 15453 The trivial group is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp

Theoremprmcyg 15454 A group with prime order is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
CycGrp

Theoremlt6abl 15455 A group with fewer than elements is abelian. (Contributed by Mario Carneiro, 27-Apr-2016.)

Theoremghmcyg 15456 The image of a cyclic group under a surjective group homomorphism is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp CycGrp

Theoremcyggex2 15457 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 15458 The exponent of a finite cyclic group is the order of the group. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx       CycGrp

Theoremcyggexb 15459 A finite abelian group is cyclic iff the exponent equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.)
gEx       CycGrp

Theoremgiccyg 15460 Cyclicity is a group property, i.e. it is preserved under isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑔 CycGrp CycGrp

Theoremcycsubgcyg 15461* The cyclic subgroup generated by is a cyclic group. (Contributed by Mario Carneiro, 24-Apr-2016.)
.g              s CycGrp

Theoremcycsubgcyg2 15462 The cyclic subgroup generated by is a cyclic group. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp       s CycGrp

10.3.3  Group sum operation

Theoremgsumval3a 15463* Value of the group sum operation over an index set with finite support. (Contributed by Mario Carneiro, 7-Dec-2014.)
Cntz                                                               g

Theoremgsumval3eu 15464* The group sum as defined in gsumval3a 15463 is uniquely defined. (Contributed by Mario Carneiro, 8-Dec-2014.)
Cntz

Theoremgsumval3 15465 Value of the group sum operation over an arbitrary finite set. (Contributed by Mario Carneiro, 15-Dec-2014.)
Cntz                                                               g

Theoremcntzcmnf 15466 Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz       CMnd

Theoremgsumcllem 15467* Lemma for gsumcl 15472 and related theorems. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)

Theoremgsumzres 15468 Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                                                 g g

Theoremgsumzcl 15469 Closure of a finite group sum. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                                          g

Theoremgsumzf1o 15470 Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                                                 g g

Theoremgsumres 15471 Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
CMnd                                   g g

Theoremgsumcl 15472 Closure of a finite group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
CMnd                            g

Theoremgsumf1o 15473 Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
CMnd                                   g g

Theoremgsumzsubmcl 15474 Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                     SubMnd                            g

Theoremgsumsubmcl 15475 Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by Mario Carneiro, 24-Apr-2016.)
CMnd              SubMnd                     g

Theoremgsumsubgcl 15476 Closure of a group sum in a subgroup. (Contributed by Mario Carneiro, 15-Dec-2014.)
SubGrp                     g

Theoremgsumzaddlem 15477* The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz                                                                             g        g g g

Theoremgsumzadd 15478 The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz                                   SubMnd                            g g g

Theoremgsumadd 15479 The sum of two group sums. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 25-Apr-2016.)
CMnd                                          g g g

Theoremgsumzsplit 15480 Split a group sum into two parts. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz                                                        g g g

Theoremgsumsplit 15481 Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.)
CMnd                                          g g g

Theoremgsumsplit2 15482* Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.)
CMnd                                          g g g

Theoremgsumconst 15483* Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
.g       g

Theoremgsumzmhm 15484 Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                            MndHom                                    g g

Theoremgsummhm 15485 Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
CMnd                     MndHom                      g g

Theoremgsummhm2 15486* Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.)
CMnd                     MndHom                             g        g

Theoremgsummulglem 15487* Lemma for gsummulg 15488 and gsummulgz 15489. (Contributed by Mario Carneiro, 7-Jan-2015.)
.g                            CMnd                     g g

Theoremgsummulg 15488* Nonnegative multiple of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 7-Jan-2015.)
.g                            CMnd              g g

Theoremgsummulgz 15489* Integer multiple of a group sum. (Contributed by Mario Carneiro, 7-Jan-2015.)
.g                                          g g

Theoremgsumzoppg 15490 The opposite of a group sum is the same as the original. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz       oppg                                          g g

Theoremgsumzinv 15491 Inverse of a group sum. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz                                                 g g

Theoremgsuminv 15492 Inverse of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 4-May-2015.)
g g

Theoremgsumsub 15493 The difference of two group sums. (Contributed by Mario Carneiro, 28-Dec-2014.)
g g g

Theoremgsumsn 15494* Group sum of a singleton. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
g

Theoremgsumunsn 15495* Append an element to a finite group sum. (Contributed by Mario Carneiro, 19-Dec-2014.)
CMnd                                                 g g

Theoremgsumpt 15496 Sum of a family that is nonzero at at most one point. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 25-Apr-2016.)
g

Theoremgsum2d 15497* Write a sum over a two-dimensional region as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                                 g g g

Theoremgsum2d2lem 15498* Lemma for gsum2d2 15499: show the function is finitely supported. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd

Theoremgsum2d2 15499* Write a group sum over a two-dimensional region as a double sum. (Note that is a function of .) (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                          g g g

Theoremgsumcom2 15500* Two-dimensional commutation of a group sum. Note that while and are constants w.r.t. , and are not. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                                        g g

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