Kernels of Homomorphisms | Abstract Algebra

We introduce kernels of homomorphisms. We give the definition of kernel in group theory, an example of a kernel of a homomorphism onto the additive group of real numbers, then finish by proving the kernel of a homomorphism is a normal subgroup of the domain group of the homomorphism. #abstractalgebra #grouptheory

Fundamental Homomorphism Theorem: (coming soon)

0:00 - Intro
0:33 - Definition of Kernel of a Homomorphism
1:26 - Example of a Kernel of a Homomorphism
5:08 - Proof Kernel is a Normal Subgroup
10:21 - Outro

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[MrCoreyTexas]

Some things were left unsaid. Fancy F(Fancy R) is a group because:. Elements would be things like 2x, x^2, 2^x, e^x, and so forth. The identity would be f(x)=0 and if you add f(x) and g(x) you get another function that's in that group. The inverse is of f(x) is simply -1*f(x), since if you add those you get 0.

So things like sin(x) would also be group elements because sin(x) + -sin(x) =0. Inverse here is different than the usual inverse you think of from high school algebra. What was throwing me off is in high school algebra, the inverse of sin() is only defined on a certain set of numbers, whereas here, in this group context, the inverse of sin(x) is just -sin(x). Math has this problem in that there are only so many words and alphabetical characters that things get reused.

[MrMegatherium]

I am just starting out. Why do we care about this object x a x inverse.... this conjugation. What does it mean? Why do we care if it is closed under conjugation any way?

[okuhlemene]

Can you please do a video of Group Action(The orbits and Stabilizers).🙏

[MrCoreyTexas]

Don't think you have showed that f(abc)=f(a)f(b)f(c) but it seems like it's not too hard to prove