Abstract Algebra | The kernel of a homomorphism

We give the definition of the kernel of a homomorphism, prove some of its properties, and give some examples.

[hyperduality]

Injective is dual to surjective synthesizes bijective or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Absolute (isomorphism, same) is dual to relative (homomorphism, similarity).
Same is dual to different.

[tmendoza]

holy hell man! just what i was looking for great vid!

[QobelD]

I bet I knew what you were thinking the moment after you said phi applied to x times phi... when you paused very slightly. .

[scrappybuilds]

Thanks so much for your instruction! Your videos have been helping me a lot this term for my abstract algebra class

[Bimallove]

You talk way too fast. This is great for those who already have a grasp of the material, but for teaching new concepts.

[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.
"Always two there are" -- Yoda.