Visual Group Theory, Lecture 4.1: Homomorphisms and isomorphisms

Visual Group Theory, Lecture 4.1: Homomorphisms and isomorphisms

A homomoprhism is function f between groups with the key property that f(ab)=f(a)f(b) holds for all elements, and an isomorphism is a bijective homomorphism. In this lecture, we use examples, Cayley diagrams, and multiplication tables to understand why this condition means that homomorphism are "structure preserving" maps. Along the way, we see some new groups, such as the roots of unity, and groups of matrices, that end up being isomorphic to groups that we familiar with.

[spyngjkru]

Thank you very much for these videos. Very useful to understand the concepts of Modern Algebra. They really helped me a lot.

[kostikvl]

37:40, slide 12/13, first branch in S3 group shall be e - (2, 3) -- (3, 1, 2) or in normalized form (1, 2, 3), now it is (1, 3, 2) that is probably typo (second branch e -- (1, 2) - (1, 3, 2) is correct)

[hyperduality]

Before (a priori, group) is dual to after (a posteriori, image) -- Immanuel Kant.
Normal subgroups are dual to homomorphism (factor groups) synthesize the kernel.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Being is dual to non-being creates becoming -- Plato.
Domains (groups) are dual to codomains (image, range).
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point, non null homotopic requires at least two points (duality) -- topology.
Polar opposites of the dyad unite into one or the monad - opposame.
Injective is dual to surjective synthesizes bijective or isomorphism.
Same is dual to different.
Isomorphism (same, absolute) is dual to homomorphism (similarity, relative).
Absolute truth is dual to relative truth -- Hume's fork.
"Always two there are" -- Yoda.

[ub]

First, Thanks for the videos!

While doodling a little thinking about the quaternions I imagined drawing their group's Cayley diagram on an extended plane, placing the identity at the origin, its opposite at the point at infinity, and the six points +/- I, J, K at the six vertices of a regular hexagon in the natural way so that I is opposite -I, etc. The result works well enough, and my question is do Cayley diagrams like this, drawn in different spaces I mean, offer any extra to the theory of groups, or can we stick to the "standard" ones and still keep everything we need?

[gayatrivenugopal]

Sir, pls include linear algebra videos tooo...

[Agus-ofrh]

Wait, at slide 12. Isn´t that the Cayley diagram for D6 instead of C6?

[scitwi]

04:25 Shouldn't it be r⁰, r¹ and r² instead of e⁰, e¹ and e²? Because the elements of the latter set are just the identity element all over again :P

[roshanshihab]

Does C6 need such a complicated Cayley graph