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Type | Label | Description |
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Statement | ||
Theorem | ga0 17001 | The action of a group on the empty set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.) |
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Theorem | gaid 17002 | 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.) |
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Theorem | subgga 17003* | A subgroup acts on its parent group. (Contributed by Jeff Hankins, 13-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.) |
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Theorem | gass 17004* | 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.) |
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Theorem | gasubg 17005 | The restriction of a group action to a subgroup is a group action. (Contributed by Mario Carneiro, 17-Jan-2015.) |
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Theorem | gaid2 17006* | 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.) |
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Theorem | galcan 17007 | The action of a particular group element is left-cancelable. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.) |
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Theorem | gacan 17008 | Group inverses cancel in a group action. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.) |
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Theorem | gapm 17009* | 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.) |
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Theorem | gaorb 17010* | 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.) |
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Theorem | gaorber 17011* | 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.) |
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Theorem | gastacl 17012* | The stabilizer subgroup in a group action. (Contributed by Mario Carneiro, 15-Jan-2015.) |
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Theorem | gastacos 17013* | Write the coset relation for the stabilizer subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.) |
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Theorem | orbstafun 17014* | Existence and uniqueness for the function of orbsta 17016. (Contributed by Mario Carneiro, 15-Jan-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
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Theorem | orbstaval 17015* | 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.) |
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Theorem | orbsta 17016* |
The Orbit-Stabilizer theorem. The mapping ![]() ![]() ![]() |
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Theorem | orbsta2 17017* | 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.) |
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Syntax | ccntz 17018 | Syntax for the centralizer of a set in a monoid. |
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Syntax | ccntr 17019 | Syntax for the centralizer of a monoid. |
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Definition | df-cntz 17020* | 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.) |
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Definition | df-cntr 17021 | Define the center of a magma, which is the elements that commute with all others. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | cntrval 17022 | Substitute definition of the center. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | cntzfval 17023* | First level substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | cntzval 17024* | Definition substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | elcntz 17025* | Elementhood in the centralizer. (Contributed by Mario Carneiro, 22-Sep-2015.) |
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Theorem | cntzel 17026* | Membership in a centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
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Theorem | cntzsnval 17027* | Special substitution for the centralizer of a singleton. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | elcntzsn 17028 | Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016.) |
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Theorem | sscntz 17029* | A centralizer expression for two sets elementwise commuting. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | cntzrcl 17030 | Reverse closure for elements of the centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
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Theorem | cntzssv 17031 | The centralizer is unconditionally a subset. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
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Theorem | cntzi 17032 | Membership in a centralizer (inference). (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.) |
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Theorem | cntri 17033 | Defining property of the center of a group. (Contributed by Mario Carneiro, 22-Sep-2015.) |
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Theorem | resscntz 17034 | Centralizer in a substructure. (Contributed by Mario Carneiro, 3-Oct-2015.) |
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Theorem | cntz2ss 17035 | Centralizers reverse the subset relation. (Contributed by Mario Carneiro, 3-Oct-2015.) |
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Theorem | cntzrec 17036 | Reciprocity relationship for centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | cntziinsn 17037* | Express any centralizer as an intersection of singleton centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | cntzsubm 17038 | Centralizers in a monoid are submonoids. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
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Theorem | cntzsubg 17039 | Centralizers in a group are subgroups. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
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Theorem | cntzidss 17040 |
If the elements of ![]() ![]() |
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Theorem | cntzmhm 17041 | Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.) |
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Theorem | cntzmhm2 17042 | Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.) |
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Theorem | cntrsubgnsg 17043 | A central subgroup is normal. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
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Theorem | cntrnsg 17044 | The center of a group is a normal subgroup. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
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Syntax | coppg 17045 | The opposite group operation. |
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Definition | df-oppg 17046 | Define an opposite group, which is the same as the original group but with addition written the other way around. df-oppr 17900 does the same thing for multiplication. (Contributed by Stefan O'Rear, 25-Aug-2015.) |
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Theorem | oppgval 17047 | 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.) |
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Theorem | oppgplusfval 17048 | Value of the addition operation of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.) |
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Theorem | oppgplus 17049 | Value of the addition operation of an opposite ring. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.) |
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Theorem | oppglem 17050 | Lemma for oppgbas 17051. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
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Theorem | oppgbas 17051 | Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
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Theorem | oppgtset 17052 | Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.) |
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Theorem | oppgtopn 17053 | Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.) |
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Theorem | oppgmnd 17054 | The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) |
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Theorem | oppgmndb 17055 | Bidirectional form of oppgmnd 17054. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
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Theorem | oppgid 17056 | Zero in a monoid is a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) |
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Theorem | oppggrp 17057 | The opposite of a group is a group. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
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Theorem | oppggrpb 17058 | Bidirectional form of oppggrp 17057. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
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Theorem | oppginv 17059 | Inverses in a group are a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
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Theorem | invoppggim 17060 | The inverse is an antiautomorphism on any group. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
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Theorem | oppggic 17061 | Every group is (naturally) isomorphic to its opposite. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
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Theorem | oppgsubm 17062 | Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
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Theorem | oppgsubg 17063 | Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
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Theorem | oppgcntz 17064 | A centralizer in a group is the same as the centralizer in the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
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Theorem | oppgcntr 17065 | The center of a group is the same as the center of the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
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Theorem | gsumwrev 17066 | 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.) |
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According to Wikipedia ("Symmetric group", 09-Mar-2019,
https://en.wikipedia.org/wiki/symmetric_group) "In abstract algebra, the
symmetric group defined over any set is the group whose elements are all the
bijections from the set to itself, and whose group operation is the composition
of functions." and according to Encyclopedia of Mathematics ("Symmetric group",
09-Mar-2019, https://www.encyclopediaofmath.org/index.php/Symmetric_group)
"The group of all permutations (self-bijections) of a set with the operation of
composition (see Permutation group).". In [Rotman] p. 27 "If X is a nonempty
set, a permutation of X is a function a : X -> X that is a one-to-one
correspondence." and "If X is a nonempty set, the symmetric group on X, denoted
SX, is the group whose elements are the permutations of X and whose
binary operation is composition of functions.". Therefore, we define the
symmetric group on a set | ||
Syntax | csymg 17067 | Extend class notation to include the class of symmetric groups. |
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Definition | df-symg 17068* |
Define the symmetric group on set ![]() ![]() |
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Theorem | symgval 17069* |
The value of the symmetric group function at ![]() |
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Theorem | symgbas 17070* | The base set of the symmetric group. (Contributed by Mario Carneiro, 12-Jan-2015.) |
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Theorem | elsymgbas2 17071 | Two ways of saying a function is a 1-1-onto mapping of A to itself. (Contributed by Mario Carneiro, 28-Jan-2015.) |
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Theorem | elsymgbas 17072 | 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.) |
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Theorem | symgbasf1o 17073 | Elements in the symmetric group are 1-1 onto functions. (Contributed by SO, 9-Jul-2018.) |
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Theorem | symgbasf 17074 | A permutation (element of the symmetric group) is a function of a set into itself. (Contributed by AV, 1-Jan-2019.) |
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Theorem | symghash 17075 |
The symmetric group on ![]() ![]() ![]() |
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Theorem | symgbasfi 17076 | The symmetric group on a finite index set is finite. (Contributed by SO, 9-Jul-2018.) |
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Theorem | symgfv 17077 | The function value of a permutation. (Contributed by AV, 1-Jan-2019.) |
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Theorem | symgfvne 17078 | The function values of a permutation for different arguments are different. (Contributed by AV, 8-Jan-2019.) |
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Theorem | symgplusg 17079* | The group operation of a symmetric group is the function composition. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.) |
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Theorem | symgov 17080 |
The value of the group operation of the symmetric group on ![]() |
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Theorem | symgcl 17081 |
The group operation of the symmetric group on ![]() |
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Theorem | symgmov1 17082* | For a permutation of a set, each element of the set replaces an(other) element of the set. (Contributed by AV, 2-Jan-2019.) |
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Theorem | symgmov2 17083* | For a permutation of a set, each element of the set is replaced by an(other) element of the set. (Contributed by AV, 2-Jan-2019.) |
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Theorem | symgbas0 17084 | The base set of the symmetric group on the empty set is the singleton containing the empty set. (Contributed by AV, 27-Feb-2019.) |
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Theorem | symg1hash 17085 |
The symmetric group on a singleton has cardinality ![]() |
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Theorem | symg1bas 17086 | The symmetric group on a singleton is the symmetric group S1 consisting of the identity only. (Contributed by AV, 9-Dec-2018.) |
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Theorem | symg2hash 17087 |
The symmetric group on a (proper) pair has cardinality ![]() |
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Theorem | symg2bas 17088 | The symmetric group on a pair is the symmetric group S2 consisting of the identity and the transposition. This theorem is also valid if the elements are identical: then it collapses to theorem symg1bas 17086. (Contributed by AV, 9-Dec-2018.) |
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Theorem | symgtset 17089 |
The topology of the symmetric group on ![]() ![]() |
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Theorem | symggrp 17090 |
The symmetric group on a set ![]() |
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Theorem | symgid 17091 |
The group identity element of the symmetric group on a set ![]() |
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Theorem | symginv 17092 | 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.) |
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Theorem | galactghm 17093* |
The currying of a group action is a group homomorphism between the group
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Theorem | lactghmga 17094* |
The converse of galactghm 17093. The uncurrying of a homomorphism into
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Theorem | symgtopn 17095 |
The topology of the symmetric group on ![]() |
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Theorem | symgga 17096* | The symmetric group induces a group action on its base set. (Contributed by Mario Carneiro, 24-Jan-2015.) |
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Theorem | pgrpsubgsymgbi 17097 | Every permutation group is a subgroup of the corresponding symmetric group. (Contributed by AV, 14-Mar-2019.) |
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Theorem | pgrpsubgsymg 17098* | Every permutation group is a subgroup of the corresponding symmetric group. (Contributed by AV, 14-Mar-2019.) |
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Theorem | idresperm 17099 | The identity function restricted to a set is a permutation of this set. (Contributed by AV, 17-Mar-2019.) |
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Theorem | idressubgsymg 17100 | The singleton containing only the identity function restricted to a set is a subgroup of the symmetric group of this set. (Contributed by AV, 17-Mar-2019.) |
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