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  • Ah, Homo morph is, um is a function between two groups that preserves the group's structure in each group.

  • It's a tool for comparing two groups for similarities.

  • Sometimes two groups are more than just similar.

  • They're identical.

  • In this case, we no longer call function a homo morph ism.

  • Instead, we call it an isil dwarfism and we say the two groups or ice Amore fink.

  • Let's give a precise definition of a nicer, more fizz.

  • Um, suppose we have two groups G and H and a home.

  • A morph is, um, f from G to h Recall that a Homo morph is, um, is a function f such that f of x times y equals f of x times.

  • Half of why now?

  • Ah, homo morph is, um, does not have to be 1 to 1.

  • It does not have to be an injection.

  • It's possible for many elements in G to map to the same element in H.

  • Similarly, F does not have to be onto.

  • It does not have to be a subjection, but for the group's G and H to be identical, we need the homo morph.

  • Isn't f to be one toe one and onto F needs to be both an injection answer.

  • Ejection.

  • This way we compare each element in G with a unique element in H and vice versa.

  • This is a definition of an ice amorphous, um, and Isil Morph is, um is a homo morph is, um that is 1 to 1.

  • And onto this can be said more briefly as an isil.

  • Morph is, um is a homo morph is, um that is also a bye Jek shin.

  • Let's now see some homo morph ISMs and determine if they're ice or morph ISMs.

  • For our first example, consider the groups of the positive real numbers under multiplication and all real numbers.

  • Under addition, one home amorphous and between them is a lager of them function.

  • We can check that this is a homo more fizz.

  • And by using the laws of logarithms, the log of X times y is equal to the log of X plus a log of y to see if this is a nice amorphous um, we have to test the fits of baiji action.

  • Let's first check that this function is 1 to 1.

  • Suppose that log of X equals log of y Since the logs are equal.

  • E to each power is equal to this simplifies to x equals y So this function is 1 to 1.

  • Next, the range of the lock function is all real numbers.

  • So this function is onto So in this example, the log function is a homo morph ism and abi ejection That makes it a nice a more fizz.

  • Um, these two groups are Isom or FIC.

  • For our next example, The first group will be the non zero complex numbers under multiplication which we denote by a C with the multiplication sign here it's understood that you are not including zero because zero does not have an inverse under multiplication.

  • The second group will be the complex numbers with absolute value of one under multiplication will denote this group by s one.

  • This is a standard notation when talking about n dimensional spheres.

  • The S is short for sphere and the one tells us that dimension in this case s one is just a circle on the complex plane with a radius of one.

  • Recall that every complex number could be written in polar form as our times e to the eighth Aita where r is the distance of a complex number two, the Origin and Veda is how far you have to rotate from the positive X axis to reach the complex number.

  • With this setup, we can now define a homo.

  • Morph is in between these two groups.

  • The function is f of our times E to the eye theta equals e to the A fatal.

  • You can visualize this by taking any non zero complex numbers e drawing away from the origin to Z and mapping Z to the point where it intersects s one.

  • But is this a home?

  • A morph ism?

  • Let's check.

  • Let ze equal are times e to the alfa and w equal s times e to the eye beta.

  • We want to check that f of z times w equals f of z times f w to begin substitute in the polar forms.

  • Next, multiply the numbers on the left using the definition of the function f we can see what each value maps too.

  • This gives us e to the alpha.

  • Plus I beta equals e to the alpha times E to thy beta.

  • This is true by the rules of exponents.

  • So this function is indeed a home, a morph ism.

  • It is also onto because for every point of the circle, there are an infinite number of complex numbers which map to it before the same reason.

  • This function is not 1 to 1.

  • So while this is a whole morph is, um, it is not a nice a morph ism.

  • Let's recap and I so more fizz.

  • Emma's a home.

  • Worf isn't that is also by ejection.

  • If you have a nice oh, Morph is in between two groups, then you say the group's heir item, or FIC.

  • This means they have the exact same group structure, even if they look different from each other.

  • The word isil dwarfism reflects the definition.

  • I so means equal and more mean shape.

  • I doubt anyone could think of a better name than this.

Ah, Homo morph is, um is a function between two groups that preserves the group's structure in each group.


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B2 中上級

同型論 (抽象代数) (Isomorphisms (Abstract Algebra))

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    林宜悉 に公開 2021 年 01 月 14 日