Tour:Factsheet two (beginners): Difference between revisions
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{{ | {{tour-factsheet| | ||
target = beginners| | |||
previous = Manipulating equations in groups| | |||
next = Confidence aggregator two (beginners)| | |||
secnum = two| | |||
previous secnum = one| | |||
next secnum = three}} | |||
To summarize some of the things we have seen so far: | To summarize some of the things we have seen so far: | ||
* The power of the structure of groups stems largely from a combination of the associativity, existence of identity element (neutral element) and the existence of inverses. | * The power of the structure of groups stems largely from a combination of the associativity, existence of identity element (neutral element) and the existence of inverses. | ||
* The uniqueness of the identity element does not require the use of associativity. However, all the other good structure of groups: including the uniqueness of inverses, | * The uniqueness of the identity element does not require the use of associativity. However, all the other good structure of groups depend on associativity: including the uniqueness of inverses, the fact that we can drop parenthesization when writing products, the fact that we can cancel elements, the fact that we can solve equations in terms of elements, the involutive nature of the inverse map, and others. | ||
* We can do many special things with finite groups, by combining the fact that there are finitely many elements, and the ability to cancel. | * We can do many special things with finite groups, by combining the fact that there are finitely many elements, and the ability to cancel. | ||
* For a subset of a finite group to be a subgroup, we only require that it be multiplicatively closed. The statement no longer remains true for infinite groups. | * For a subset of a finite group to be a subgroup, we only require that it be multiplicatively closed. The statement no longer remains true for infinite groups. | ||
* The left quotient and right quotient expressions can be used to test whether a subset is a subgroup. | |||
{{tour-factsheet| | |||
target = beginners| | |||
previous = Manipulating equations in groups| | |||
next = Confidence aggregator two (beginners)| | |||
secnum = two| | |||
previous secnum = one| | |||
next secnum = three}} | |||
Latest revision as of 19:45, 31 March 2022
This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
PREVIOUS: Manipulating equations in groups| UP: Introduction two | NEXT: Confidence aggregator two (beginners)
PREVIOUS SECTION FACTSHEET: Factsheet one|NEXT SECTION FACTSHEET: Factsheet three
General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part
To summarize some of the things we have seen so far:
- The power of the structure of groups stems largely from a combination of the associativity, existence of identity element (neutral element) and the existence of inverses.
- The uniqueness of the identity element does not require the use of associativity. However, all the other good structure of groups depend on associativity: including the uniqueness of inverses, the fact that we can drop parenthesization when writing products, the fact that we can cancel elements, the fact that we can solve equations in terms of elements, the involutive nature of the inverse map, and others.
- We can do many special things with finite groups, by combining the fact that there are finitely many elements, and the ability to cancel.
- For a subset of a finite group to be a subgroup, we only require that it be multiplicatively closed. The statement no longer remains true for infinite groups.
- The left quotient and right quotient expressions can be used to test whether a subset is a subgroup.
This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
PREVIOUS: Manipulating equations in groups| UP: Introduction two | NEXT: Confidence aggregator two (beginners)
PREVIOUS SECTION FACTSHEET: Factsheet one|NEXT SECTION FACTSHEET: Factsheet three
General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part