When studying algebraic structures, we are interested in the maps between them that are "compatible" with their structure. In the case of groups, such maps are called homomorphisms, and here we introduce them, their basic properties and many examples. We go on to study the kernel of a group homomorphism, which is the subgroup that is mapped to the neutral element. It is an important tool in studying homomorphisms and groups. In the last part, we discover that the automorphisms of a group (bijective homomorphisms from the group to itself) form a group themselves.
Sections
00:00:00 Greetings
00:00:42 1. Homomorphisms of groups
00:03:30 Definition of homomorphism, isomorphism, endomorphism, automorphism
00:09:22 First properties and examples of homomorphisms
00:39:17 2. The kernel
00:39:28 Definition of the kernel of a homomorphism
00:40:02 Kernel theorem
00:48:36 Cosets of the kernel
01:00:43 Modular arithmetic as an example
01:09:20 3. Automorphism groups
01:16:22 Inner autormorphisms (conjugations)
01:30:46 The center of a group
This video was made using LaTeX Beamer, OmniGraffle, LaTeXit and SlidePilot.
#maths #algebra #groups #grouptheory #subgroups #symmetry #permutations #integers #homomorphism #kernels #automorphism
#modulararithmetic #isomorphism