S. D. V. Stephens

notes / Studies in Algebra and Group Theory / chapter 1: Sets and Groups / From my notebooks

N8 Lecture 1 and 2 Review: Functions, Cardinality and Groups

Lecture 1 review

Definition N8.1 (Injective, surjective, bijective). A function is

  • injective (one-to-one, ) implies ;
  • surjective (onto, ) for all there exists such that ;
  • bijective (one-to-one correspondence) injective and surjective.

Main definitions

Definition N8.2 (Groups, monoids, semigroups).

  • Group: a set with a binary operation given by some operation (, etc.), with identity (), inverses () and associativity.
  • Monoid: the same, sans inverses.
  • Semigroup: the same, sans identity.
  • Abelian: the same, with commutativity.

Definition N8.3 (Isomorphism). and have the same group structure.

Definition N8.4 (Symmetric group). Given elements, .

Definition N8.5 (Permutation groups). : the bijections , with composition ().

Definition N8.6 (Product groups). with . Order: , where , .

Lecture 2 review

Definition N8.7 (Cardinality). and have the same cardinality there is a bijection . Also

  • there exists a surjection ;
  • there exists an injection .

Theorem N8.8 (Schröder–Bernstein). If there exist injections and , then there exists a bijection , so .

TRY TO PROVE:

Scratch work (no finished proof on the page):

  • Maps and : for every there should exist such that ; , .
  • Arrays of numbers:
2 3 5 7 11 13 17
1 3 5 7 11 13 17
2 6 10 14 22 26 34
4 12 20
8

and
1 3 5 7 9 11
2 6 10 14 18 22
4
8
  • , and .
  • ; ; , :
  • , , , …
  • , , , , , .