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Theorem List for Metamath Proof Explorer - 15001-15100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlagsubg2 15001 Lagrange's theorem for finite groups. Call the "order" of a group the cardinal number of the basic set of the group, and "index of a subgroup" the cardinal number of the set of left (or right, this is the same) cosets of this subgroup. Then the order of the group is the (cardinal) product of the order of any of its subgroups by the index of this subgroup. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
~QG        SubGrp

Theoremlagsubg 15002 Lagrange theorem for Groups: the order of any subgroup of a finite group is a divisor of the order of the group. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
SubGrp

10.2.3  Elementary theory of group homomorphisms

Syntaxcghm 15003 Extend class notation with the generator of group hom-sets.

Definitiondf-ghm 15004* A homomorphism of groups is a map between two structures which preserves the group operation. Requiring both sides to be groups simplifies most theorems at the cost of complicating the theorem which pushes forward a group structure. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremreldmghm 15005 Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremisghm 15006* Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremisghm3 15007* Property of a group homomorphism, similar to ismhm 14740. (Contributed by Mario Carneiro, 7-Mar-2015.)

Theoremghmgrp1 15008 A group homomorphism is only defined when the domain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghmgrp2 15009 A group homomorphism is only defined when the codomain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghmf 15010 A group homomorphism is a function. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghmlin 15011 A homomorphism of groups is linear. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghmid 15012 A homomorphism of groups preserves the identity. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghminv 15013 A homomorphism of groups preserves inverses. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghmsub 15014 Linearity of subtraction through a group homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremisghmd 15015* Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)

Theoremghmmhm 15016 A group homorphism is a monoid homorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
MndHom

Theoremghmmhmb 15017 Group homorphisms and monoid homomorphisms coincide. (Thus, is somewhat redundant, although its stronger reverse closure properties are sometimes useful.) (Contributed by Stefan O'Rear, 7-Mar-2015.)
MndHom

Theoremghmmulg 15018 A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
.g       .g

Theoremghmrn 15019 The range of a homomorphism is a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
SubGrp

Theorem0ghm 15020 The constant zero linear function between two groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremidghm 15021 The identity homomorphism on a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremresghm 15022 Restriction of a homomorphism to a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
s        SubGrp

Theoremresghm2 15023 One direction of resghm2b 15024. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
s        SubGrp

Theoremresghm2b 15024 Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
s        SubGrp

Theoremghmco 15025 The composition of group homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)

Theoremghmima 15026 The image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
SubGrp SubGrp

Theoremghmpreima 15027 The inverse image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
SubGrp SubGrp

Theoremghmeql 15028 The equalizer of two group homomorphisms is a subgroup. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
SubGrp

Theoremghmnsgima 15029 The image of a normal subgroup under a surjective homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
NrmSGrp NrmSGrp

Theoremghmnsgpreima 15030 The inverse image of a normal subgroup under a homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
NrmSGrp NrmSGrp

Theoremghmker 15031 The kernel of a homomorphism is a normal subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)
NrmSGrp

Theoremghmeqker 15032 Two source points map to the same destination point under a group homomorphism iff their difference belongs to the kernel. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theorempwsdiagghm 15033* Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s

Theoremghmf1 15034* Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremghmf1o 15035 A bijective group homomorphism is an isomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.)

Theoremconjghm 15036* Conjugation is an automorphism of the group. (Contributed by Mario Carneiro, 13-Jan-2015.)

Theoremconjsubg 15037* A conjugated subgroup is also a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
SubGrp SubGrp

Theoremconjsubgen 15038* A conjugated subgroup is equinumerous to the original subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
SubGrp

Theoremconjnmz 15039* A subgroup is unchanged under conjugation by an element of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
SubGrp

Theoremconjnmzb 15040* Alternative condition for elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
SubGrp

Theoremconjnsg 15041* A normal subgroup is unchanged under conjugation. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp

Theoremdivsghm 15042* If is a normal subgroup of , then the "natural map" from elements to their cosets is a group homomorphism from to . (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 18-Sep-2015.)
s ~QG        ~QG        NrmSGrp

Theoremghmpropd 15043* Group homomorphism depends only on the group attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)

10.2.4  Isomorphisms of groups

Syntaxcgim 15044 The class of group isomorphism sets.
GrpIso

Syntaxcgic 15045 The class of the group isomorphism relation.
𝑔

Definitiondf-gim 15046* An isomorphism of groups is a homomorphism which is also a bijection, i.e. it preserves equality as well as the group operation. (Contributed by Stefan O'Rear, 21-Jan-2015.)
GrpIso

Definitiondf-gic 15047 Two groups are said to be isomorphic iff they are connected by at least one isomorphism. Isomophic groups share all global group properties, but to relate local properties requires knowledge of a specific isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑔 GrpIso

Theoremgimfn 15048 The group isomorphism function is a well-defined function. (Contributed by Mario Carneiro, 23-Aug-2015.)
GrpIso

Theoremisgim 15049 An isomorphism of groups is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
GrpIso

Theoremgimf1o 15050 An isomorphism of groups is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
GrpIso

Theoremgimghm 15051 An isomorphism of groups is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
GrpIso

Theoremisgim2 15052 A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 17791. (Contributed by Mario Carneiro, 6-May-2015.)
GrpIso

Theoremsubggim 15053 Behavior of subgroups under isomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
GrpIso SubGrp SubGrp

Theoremgimcnv 15054 The converse of a bijective group homomorphism is a bijective group homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
GrpIso GrpIso

Theoremgimco 15055 The composition of group isomorphisms is a group isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
GrpIso GrpIso GrpIso

Theorembrgic 15056 The relation "is isomorphic to" for groups. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑔 GrpIso

Theorembrgici 15057 Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
GrpIso 𝑔

Theoremgicref 15058 Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑔

Theoremgiclcl 15059 Isomorphism implies the left side is a group. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑔

Theoremgicrcl 15060 Isomorphism implies the right side is a group. (Contributed by Mario Carneiro, 6-May-2015.)
𝑔

Theoremgicsym 15061 Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑔 𝑔

Theoremgictr 15062 Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑔 𝑔 𝑔

Theoremgicer 15063 Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑔

Theoremgicen 15064 Isomorphic groups have equinumerous base sets. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑔

Theoremgicsubgen 15065 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 15066 Extend class definition to include the class of group actions.

Definitiondf-ga 15067* 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 15068* 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 15083). 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 15069 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 15070 The right argument of a group action is a set. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremgagrpid 15071 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 15072 The mapping of the group action operation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremgafo 15073 A group action is onto its base set. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremgaass 15074 An "associative" property for group actions. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremga0 15075 The action of a group on the empty set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremgaid 15076 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 15077* 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 15078* 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 15079 The restriction of a group action to a subgroup is a group action. (Contributed by Mario Carneiro, 17-Jan-2015.)
s        SubGrp

Theoremgaid2 15080* 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 15081 The action of a particular group element is left-cancelable. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremgacan 15082 Group inverses cancel in a group action. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Theoremgapm 15083* 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 15084* 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 15085* 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 15086* The stabilizer subgroup in a group action. (Contributed by Mario Carneiro, 15-Jan-2015.)
SubGrp

Theoremgastacos 15087* Write the coset relation for the stabilizer subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
~QG

Theoremorbstafun 15088* Existence and uniqueness for the function of orbsta 15090. (Contributed by Mario Carneiro, 15-Jan-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
~QG

Theoremorbstaval 15089* 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 15090* 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 15091* 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 15092 Extend class notation to include the class of symmetry groups.

Definitiondf-symg 15093* 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 15094* The value of the symmetry group function at . (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.)
TopSet

Theoremsymgbas 15095* The base set of the symmetric group. (Contributed by Mario Carneiro, 12-Jan-2015.)

Theoremelsymgbas2 15096 Two ways of saying a function is a 1-1-onto mapping of A to itself. (Contributed by Mario Carneiro, 28-Jan-2015.)

Theoremelsymgbas 15097 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 15098 The symmetric group on objects has cardinality . (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremsymgplusg 15099* The value of the symmetry group function at . (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.)

Theoremsymgov 15100 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.)

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