Visual Group Theory, Lecture 4.1: Homomorphisms and isomorphisms

[@spyngjkru6095]

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)

[@hyperduality2838]

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.

[@hypercube79]

thank you Sir.

[@Adityarm.08]

Thank you so much!!

[@u2b84]

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?

[@gayatrivenugopal1577]

Sir, pls include linear algebra videos tooo...

[@Agus-of6rh]

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

[@scitwi9164]

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

[@seanki98]

Correct, and it says as such on the slides, I think he just misspoke

[@roshanshihab8515]

Does C6 need such a complicated Cayley graph

[@Mrpallekuling]

No, it can look just like Z6, but at 34:30 the diagram is more complex to illustrate that it can be difficult to identify an isomorphism