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

Theoremgicer 15101 Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑔

Theoremgicen 15102 Isomorphic groups have equinumerous base sets. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑔

Theoremgicsubgen 15103 A less trivial example of a group invariant: cardinality of the subgroup lattice. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑔 SubGrp SubGrp

10.2.5  Group actions

Syntaxcga 15104 Extend class definition to include the class of group actions.

Definitiondf-ga 15105* Define the class of all group actions. A group acts on a set if a permutation on is associated with every element of in such a way that the identity permutation on is associated with the neutral element of , and the composition of the permutations associated with two elements of is identical with the permutation associated to the composition of these two elements (in the same order) in the group . (Contributed by Jeff Hankins, 10-Aug-2009.)

Theoremisga 15106* The predicate "is a (left) group action." The group is said to act on the base set of the action, which is not assumed to have any special properties. There is a related notion of right group action, but as the Wikipedia article explains, it is not mathematically interesting. The way actions are usually thought of is that each element of is a permutation of the elements of (see gapm 15121). Since group theory was classically about symmetry groups, it is therefore likely that the notion of group action was useful even in early group theory. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremgagrp 15107 The left argument of a group action is a group. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremgaset 15108 The right argument of a group action is a set. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremgagrpid 15109 The identity of the group does not alter the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremgaf 15110 The mapping of the group action operation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremgafo 15111 A group action is onto its base set. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremgaass 15112 An "associative" property for group actions. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremga0 15113 The action of a group on the empty set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremgaid 15114 The trivial action of a group on any set. Each group element corresponds to the identity permutation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)

Theoremsubgga 15115* A subgroup acts on its parent group. (Contributed by Jeff Hankins, 13-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
s               SubGrp

Theoremgass 15116* A subset of a group action is a group action iff it is closed under the group action operation. (Contributed by Mario Carneiro, 17-Jan-2015.)

Theoremgasubg 15117 The restriction of a group action to a subgroup is a group action. (Contributed by Mario Carneiro, 17-Jan-2015.)
s        SubGrp

Theoremgaid2 15118* A group operation is a left group action of the group on itself. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremgalcan 15119 The action of a particular group element is left-cancelable. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremgacan 15120 Group inverses cancel in a group action. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremgapm 15121* The action of a particular group element is a permutation of the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)

Theoremgaorb 15122* The orbit equivalence relation puts two points in the group action in the same equivalence class iff there is a group element that takes one element to the other. (Contributed by Mario Carneiro, 14-Jan-2015.)

Theoremgaorber 15123* The orbit equivalence relation is an equivalence relation on the target set of the group action. (Contributed by NM, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremgastacl 15124* The stabilizer subgroup in a group action. (Contributed by Mario Carneiro, 15-Jan-2015.)
SubGrp

Theoremgastacos 15125* Write the coset relation for the stabilizer subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
~QG

Theoremorbstafun 15126* Existence and uniqueness for the function of orbsta 15128. (Contributed by Mario Carneiro, 15-Jan-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
~QG

Theoremorbstaval 15127* Value of the function at a given equivalence class element. (Contributed by Mario Carneiro, 15-Jan-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
~QG

Theoremorbsta 15128* The Orbit-Stabilizer theorem. The mapping is a bijection from the cosets of the stabilizer subgroup of to the orbit of . (Contributed by Mario Carneiro, 15-Jan-2015.)
~QG

Theoremorbsta2 15129* Relation between the size of the orbit and the size of the stabilizer of a point in a finite group action. (Contributed by Mario Carneiro, 16-Jan-2015.)
~QG

10.2.6  Symmetry groups and Cayley's Theorem

Syntaxcsymg 15130 Extend class notation to include the class of symmetry groups.

Definitiondf-symg 15131* Define the symmetry group on set . We represent the group as the set of 1-1-onto functions from to itself under function composition, and topologize it as a function space assuming the set is discrete. (Contributed by Paul Chapman, 25-Feb-2008.)
TopSet

Theoremsymgval 15132* The value of the symmetry group function at . (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.)
TopSet

Theoremsymgbas 15133* The base set of the symmetric group. (Contributed by Mario Carneiro, 12-Jan-2015.)

Theoremelsymgbas2 15134 Two ways of saying a function is a 1-1-onto mapping of A to itself. (Contributed by Mario Carneiro, 28-Jan-2015.)

Theoremelsymgbas 15135 Two ways of saying a function is a 1-1-onto mapping of A to itself. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.)

Theoremsymghash 15136 The symmetric group on objects has cardinality . (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremsymgplusg 15137* The value of the symmetry group function at . (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.)

Theoremsymgov 15138 The value of the group operation of the symmetry group on . (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.)

Theoremsymgcl 15139 The group operation of the symmetry group on is closed, i.e. a magma. (Contributed by Mario Carneiro, 12-Jan-2015.) (Revised by Mario Carneiro, 28-Jan-2015.)

Theoremsymgtset 15140 The topology of the symmetry group on . This component is defined on a larger set than the true base - the product topology is defined on the set of all functions, not just bijections - but the definition of ensures that it is trimmed down before it gets use. (Contributed by Mario Carneiro, 29-Aug-2015.)
TopSet

Theoremsymggrp 15141 The symmetry group on is a group. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremsymgid 15142 The value of the identity element of the symmetry group on (Contributed by Paul Chapman, 25-Jul-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremsymginv 15143 The group inverse in the symmetric group corresponds to the functional inverse. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)

Theoremgalactghm 15144* The currying of a group action is a group homomorphism between the group and the symetry group . (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)

Theoremlactghmga 15145* The converse of galactghm 15144. The uncurrying of a homomorphism into is a group action. Thus, group actions and group homomorphisms into a symmetric group are essentially equivalent notions. (Contributed by Mario Carneiro, 15-Jan-2015.)

Theoremsymgtopn 15146 The topology of the symmetry group on . (Contributed by Mario Carneiro, 29-Aug-2015.)
t

Theoremsymgga 15147* The symmetric group induces a group action on its base set. (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremcayleylem1 15148* Lemma for cayley 15150. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremcayleylem2 15149* Lemma for cayley 15150. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremcayley 15150* Cayley's Theorem (constructive version): given group , is an isomorphism between and the subgroup of the symmetry group on the underlying set of . (Contributed by Paul Chapman, 3-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
SubGrp s

Theoremcayleyth 15151* Cayley's Theorem (existence version): every group is isomorphic to a subgroup of the symmetry group on the underlying set of . (For any group there exists an isomorphism between and a subgroup of the symmetry group on the underlying set of .) (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
SubGrp s

10.2.7  Centralizers and centers

Syntaxccntz 15152 Syntax for the centralizer of a set in a monoid.
Cntz

Syntaxccntr 15153 Syntax for the centralizer of a monoid.
Cntr

Definitiondf-cntz 15154* Define the centralizer of a subset of a magma, which is the set of elements each of which commutes with each element of the given subset. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Cntz

Definitiondf-cntr 15155 Define the center of a magma, which is the elements that commute with all others. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Cntr Cntz

Theoremcntrval 15156 Substitute definition of the center. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Cntz       Cntr

Theoremcntzfval 15157* First level substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Cntz

Theoremcntzval 15158* Definition substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Cntz

Theoremelcntz 15159* Elementhood in the centralizer. (Contributed by Mario Carneiro, 22-Sep-2015.)
Cntz

Theoremcntzel 15160* Membership in a centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Cntz

Theoremcntzsnval 15161* Special substitution for the centralizer of a singleton. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Cntz

Theoremelcntzsn 15162 Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz

Theoremsscntz 15163* A centralizer expression for two sets elementwise commuting. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Cntz

Theoremcntzrcl 15164 Reverse closure for elements of the centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Cntz

Theoremcntzssv 15165 The centralizer is unconditionally a subset. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Cntz

Theoremcntzi 15166 Membership in a centralizer (inference). (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
Cntz

Theoremcntri 15167 Defining property of the center of a group. (Contributed by Mario Carneiro, 22-Sep-2015.)
Cntr

Theoremresscntz 15168 Centralizer in a substructure. (Contributed by Mario Carneiro, 3-Oct-2015.)
s        Cntz       Cntz

Theoremcntz2ss 15169 Centralizers reverse the subset relation. (Contributed by Mario Carneiro, 3-Oct-2015.)
Cntz

Theoremcntzrec 15170 Reciprocity relationship for centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Cntz

Theoremcntziinsn 15171* Express any centralizer as an intersection of singleton centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Cntz

Theoremcntzsubm 15172 Centralizers in a monoid are submonoids. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
Cntz       SubMnd

Theoremcntzsubg 15173 Centralizers in a group are subgroups. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Cntz       SubGrp

Theoremcntzidss 15174 If the elements of commute, the elements of a subset also commute. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz

Theoremcntzmhm 15175 Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz       Cntz       MndHom

Theoremcntzmhm2 15176 Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz       Cntz       MndHom

Theoremcntrsubgnsg 15177 A central subgroup is normal. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Cntr       SubGrp NrmSGrp

Theoremcntrnsg 15178 The center of a group is a normal subgroup. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Cntr       NrmSGrp

10.2.8  The opposite group

Syntaxcoppg 15179 The opposite group operation.
oppg

Definitiondf-oppg 15180 Define an opposite group, which is the same as the original group but with addition written the other way around. df-oppr 15766 does the same thing for multiplication. (Contributed by Stefan O'Rear, 25-Aug-2015.)
oppg sSet tpos

Theoremoppgval 15181 Value of the opposite group. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
oppg       sSet tpos

Theoremoppgplusfval 15182 Value of the addition operation of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
oppg              tpos

Theoremoppgplus 15183 Value of the addition operation of an opposite ring. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
oppg

Theoremoppglem 15184 Lemma for oppgbas 15185. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg       Slot

Theoremoppgbas 15185 Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg

Theoremoppgtset 15186 Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
oppg       TopSet       TopSet

Theoremoppgtopn 15187 Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
oppg

Theoremoppgmnd 15188 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 15189 Bidirectional form of oppgmnd 15188. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg

Theoremoppgid 15190 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 15191 The opposite of a group is a group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg

Theoremoppggrpb 15192 Bidirectional form of oppggrp 15191. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg

Theoremoppginv 15193 Inverses in a group are a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg

Theoreminvoppggim 15194 The inverse is an antiautomorphism on any group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg              GrpIso

Theoremoppggic 15195 Every group is (naturally) isomorphic to its opposite. (Contributed by Stefan O'Rear, 26-Aug-2015.)
oppg       𝑔

Theoremoppgsubm 15196 Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
oppg       SubMnd SubMnd

Theoremoppgsubg 15197 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
oppg       SubGrp SubGrp

Theoremoppgcntz 15198 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 15199 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 15200 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

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