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Theorem List for Metamath Proof Explorer - 14601-14700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsubmmulgcl 14601 Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 13-Jan-2015.)
.g       SubMnd

Theoremsubmmulg 14602 A group multiple is the same if evaluated in a submonoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
.g       s        .g       SubMnd

Theoremprdsinvlem 14603* Characterization of inverses in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
s

Theoremprdsgrpd 14604 The product of a family of groups is a group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theoremprdsinvgd 14605* Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theorempwsgrp 14606 The product of a family of groups is a group. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Theorempwsinvg 14607 Negation in a group power. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Theorempwssub 14608 Subtraction in a group power. (Contributed by Mario Carneiro, 12-Jan-2015.)
s

Theorempwsmulg 14609 Value of a group multiple in a structure power. (Contributed by Mario Carneiro, 15-Jun-2015.)
s               .g       .g

Theoremimasgrp2 14610* The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
s

Theoremimasgrp 14611* The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
s

Theoremimasgrpf1 14612 The image of a group under an injection is a group. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

Theoremdivsgrp2 14613* Prove that a quotient structure is a group. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
s

Theoremxpsgrp 14614 The binary product of groups is a group. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

10.2.2  Subgroups and Quotient groups

Syntaxcsubg 14615 Extend class notation with all subgroups of a group.
SubGrp

Syntaxcnsg 14616 Extend class notation with all normal subgroups of a group.
NrmSGrp

Syntaxcqg 14617 Quotient group equivalence class.
~QG

Definitiondf-subg 14618* Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 14636), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14631), contains the neutral element of the group (see subg0 14627) and contains the inverses for all of its elements (see subginvcl 14630). (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp s

Definitiondf-nsg 14619* Define the equivalence relation in a quotient ring or quotient group (where is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)
NrmSGrp SubGrp

Definitiondf-eqg 14620* Define the equivalence relation in a quotient ring or quotient group (where is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)
~QG

Theoremissubg 14621 The subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp s

Theoremsubgss 14622 A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremsubgid 14623 A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.)
SubGrp

Theoremsubggrp 14624 A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.)
s        SubGrp

Theoremsubgbas 14625 The base of the restricted group in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
s        SubGrp

Theoremsubgrcl 14626 Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremsubg0 14627 A subgroup of a group must have the same identity as the group. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
s               SubGrp

Theoremsubginv 14628 The inverse of an element in a subgroup is the same as the inverse in the larger group. (Contributed by Mario Carneiro, 2-Dec-2014.)
s                      SubGrp

Theoremsubg0cl 14629 The group identity is an element of any subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremsubginvcl 14630 The inverse of an element is closed in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremsubgcl 14631 A subgroup is closed under group operation. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremsubgsubcl 14632 A subgroup is closed under group subtraction. (Contributed by Mario Carneiro, 18-Jan-2015.)
SubGrp

Theoremsubgsub 14633 The subtraction of elements in a subgroup is the same as subtraction in the group. (Contributed by Mario Carneiro, 15-Jun-2015.)
s               SubGrp

Theoremsubgmulgcl 14634 Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
.g       SubGrp

Theoremsubgmulg 14635 A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
.g       s        .g       SubGrp

Theoremissubg2 14636* Characterize the subgroups of a group by closure properties. (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp

Theoremissubg3 14637* A subgroup is a symmetric submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubGrp SubMnd

Theoremissubg4 14638* A subgroup is a nonempty subset of the group closed under subtraction. (Contributed by Mario Carneiro, 17-Sep-2015.)
SubGrp

Theoremsubgsubm 14639 A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.)
SubGrp SubMnd

Theoremsubsubg 14640 A subgroup of a subgroup is a subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)
s        SubGrp SubGrp SubGrp

Theoremsubgint 14641 The intersection of a nonempty collection of subgroups is a subgroup. (Contributed by Mario Carneiro, 7-Dec-2014.)
SubGrp SubGrp

Theorem0subg 14642 The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.)
SubGrp

Theoremcycsubgcl 14643* The set of integer powers of an element of a group forms a subgroup containing , called the cyclic group generated by the element . (Contributed by Mario Carneiro, 13-Jan-2015.)
.g              SubGrp

Theoremcycsubgss 14644* The cyclic subgroup generated by an element is a subset of any subgroup containing . (Contributed by Mario Carneiro, 13-Jan-2015.)
.g              SubGrp

Theoremcycsubg 14645* The cyclic group generated by is the smallest subgroup containing . (Contributed by Mario Carneiro, 13-Jan-2015.)
.g              SubGrp

Theoremisnsg 14646* Property of being a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp SubGrp

Theoremisnsg2 14647* Weaken the condition of isnsg 14646 to only one side of the implication. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp SubGrp

Theoremnsgbi 14648 Defining property of a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp

Theoremnsgsubg 14649 A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp SubGrp

Theoremnsgconj 14650 The conjugation of an element of a normal subgroup is in the subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)
NrmSGrp

Theoremisnsg3 14651* A subgroup is normal iff the conjugation of all the elements of the subgroup is in the subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp SubGrp

Theoremsubgacs 14652 Subgroups are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
SubGrp ACS

Theoremnsgacs 14653 Normal subgroups form an algebraic closure system. (Contributed by Stefan O'Rear, 4-Sep-2015.)
NrmSGrp ACS

Theoremcycsubg2 14654* The subgroup generated by an element is exhausted by its multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.)
.g              mrClsSubGrp

Theoremcycsubg2cl 14655 Any multiple of an element is contained in the generated cyclic subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
.g       mrClsSubGrp

Theoremelnmz 14656* Elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)

Theoremnmzbi 14657* Defining property of the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)

Theoremnmzsubg 14658* The normalizer NG(S) of a subset of the group is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
SubGrp

Theoremssnmz 14659* A subgroup is a subset of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
SubGrp

Theoremisnsg4 14660* A subgroup is normal iff its normalizer is the entire group. (Contributed by Mario Carneiro, 18-Jan-2015.)
NrmSGrp SubGrp

Theoremnmznsg 14661* Any subgroup is a normal subgroup of its normalizer. (Contributed by Mario Carneiro, 19-Jan-2015.)
s        SubGrp NrmSGrp

Theorem0nsg 14662 The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
NrmSGrp

Theoremnsgid 14663 The whole group is a normal subgroup of itself. (Contributed by Mario Carneiro, 4-Feb-2015.)
NrmSGrp

Theoremreleqg 14664 The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
~QG

Theoremeqgfval 14665* Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.)
~QG

Theoremeqgval 14666 Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
~QG

Theoremeqger 14667 The subgroup coset equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 13-Jan-2015.)
~QG        SubGrp

Theoremeqglact 14668* A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)
~QG

Theoremeqgid 14669 The left coset containing the identity is the original subgroup. (Contributed by Mario Carneiro, 20-Sep-2015.)
~QG               SubGrp

Theoremeqgen 14670 Each coset is equipotent to the subgroup itself (which is also the coset containing the identity). (Contributed by Mario Carneiro, 20-Sep-2015.)
~QG        SubGrp

Theoremeqgcpbl 14671 The subgroup coset equivalence relation is compatible with addition when the subgroup is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
~QG               NrmSGrp

Theoremdivsgrp 14672 If is a normal subgroup of , then is a group, called the quotient of by . (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
s ~QG        NrmSGrp

Theoremdivseccl 14673 Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                      NrmSGrp ~QG

Theoremdivsadd 14674 Value of the group operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             NrmSGrp ~QG ~QG ~QG

Theoremdivs0 14675 Value of the group identity operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG               NrmSGrp ~QG

Theoremdivsinv 14676 Value of the group inverse operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             NrmSGrp ~QG ~QG

Theoremdivssub 14677 Value of the group subtraction operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             NrmSGrp ~QG ~QG ~QG

Theoremlagsubg2 14678 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 14679 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 14680 Extend class notation with the generator of group hom-sets.

Definitiondf-ghm 14681* 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 14682 Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremisghm 14683* Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremisghm3 14684* Property of a group homomorphism, similar to ismhm 14417. (Contributed by Mario Carneiro, 7-Mar-2015.)

Theoremghmgrp1 14685 A group homomorphism is only defined when the domain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghmgrp2 14686 A group homomorphism is only defined when the codomain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghmf 14687 A group homomorphism is a function. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghmlin 14688 A homomorphism of groups is linear. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghmid 14689 A homomorphism of groups preserves the identity. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghminv 14690 A homomorphism of groups preserves inverses. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremghmsub 14691 Linearity of subtraction through a group homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremisghmd 14692* Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)

Theoremghmmhm 14693 A group homorphism is a monoid homorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
MndHom

Theoremghmmhmb 14694 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 14695 A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
.g       .g

Theoremghmrn 14696 The range of a homomorphism is a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
SubGrp

Theorem0ghm 14697 The constant zero linear function between two groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremidghm 14698 The identity homomorphism on a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Theoremresghm 14699 Restriction of a homomorphism to a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
s        SubGrp

Theoremresghm2 14700 One direction of resghm2b 14701. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
s        SubGrp

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