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.