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  • Imagine, if you will, that we have two groups which we will creatively callGroup 1”

  • andGroup 2.” How would you compare these two groups? That is, how would you determine

  • how similar or different they are from each other? And what do we mean bysimilar”?

  • What features of groups are we even using to compare them? To answer these questions

  • and more, we will use a tool called a “homomorphism.”

  • Consider the group of integers “Z”, and

  • the integers mod 2. The operation well use in both groups is

  • addition. The integers are an infinite group, while

  • the integers mod 2 is a finite group with only 2 elements.

  • So at first you might think these two groups are completely different from each other.

  • But consider this. The integers can be broken into two sets: the even and odd integers.

  • If you add one even integer with another, you get an even integer.

  • If you add an even and an odd, you get an odd.

  • If you add an odd and an even, you get an odd.

  • And finally, if you add two odds, you get an even integer.

  • Now compare this with the integers mod 2. In this group, 0 plus 0 equals 0.

  • 0 + 1 equals 1. 1 + 0 equals 1.

  • And 1 + 1 equals 0. Do you notice a similarity?

  • If you replaceEvenwith 0, andOddwith 1, then these say the exact same thing.

  • This an interesting observation. By splitting the integers into two sets - evens and odds

  • - we see these two sets behaved exactly the same as the group of integers mod 2. We can

  • express this observation in mathematical terms by defining a function “F” from the integers

  • to the integers mod two. The function is simple: if an integer is even, map it to zero. If

  • an integer is odd, send it to one.

  • Is it possible to create a reverse function that encodes our observation that evens and

  • odds behave like the group of integers mod 2?

  • The answer isno. Suppose we tried to create a function G from

  • the integers mod 2 to the integers. Well send “0 mod 2” to some integer

  • “x”, and “1 mod 2” to some integer “y”.

  • Since “0 + 0 = 0 mod 2”, we want “x + x = x” in the integers.

  • This is because if we are comparing groups, then the behavior of elements in the output

  • should be similar to the behavior of the inputs. Solving the equation “x + x = x” gives

  • us “x = 0”. So G maps “0 mod 2” to 0.

  • Next, “1 + 1 = 0 mod 2”, so we want “y + y = 0” in the integers.

  • Again, this is because we want the inputs and outputs to behave similarly.

  • If we solve “y + y = 0” we get “y = 0” So G also maps “1 mod 2” to 0…

  • The function G is rather trivial. It sends everything in the

  • integers mod 2” to 0… In this direction, we lose the group similarity between the evens

  • and odds and theintegers mod 2.” Keep this in mind when using a function to compare

  • two groups. The direction can matter.

  • Let’s see another example where we take two groups and compare them for similarities.

  • The integers mod 4, with addition as the group operationand the group of four numbers

  • 1.. -1.. i.. and -i with multiplication as the operation. To compare them, let’s look

  • at the Cayley table for both groups.

  • First, the group of integers mod 4 with the

  • operation of addition. The way you say this aloud isthe group

  • of integers mod 4 under addition.” This is the jargon used in math for specifying

  • the group operation. In the upper left corner, write the group

  • operation. Then in the first row and first column, list

  • all the elements in the group. Next, we go through and compute all possible

  • 16 operations. 0 + 0 = 0 mod 4…

  • 0 + 1 = 1 mod 4…

  • 0 + 2 = 2 mod 4…

  • 0 + 3 = 3 mod 4…

  • Continuing, 1 + 0 = 1…

  • 1 + 1 = 2…

  • 1 + 2 = 3…

  • and 1 + 3 = 0. And well just go ahead and fill out the

  • remaining 8 squares

  • Next, let’s make the Cayley table for the

  • numbers 1, -1, i and -i under multiplication.

  • To finish, we quickly perform the 16 multiplications.

  • We now have the Cayley tables for these two groups. Theyre both the same size - they

  • each have 4 elements. But do they behave similarly? Or are they different and unrelated to each

  • other? Well answer this question visually by coloring the squares.

  • Each group has an identity element.

  • For the integers mod 4 it’s the number 0. And for the second group it’s the number

  • 1. Let’s highlight the identity elements red.

  • To help you see the pattern, let’s switch “-1” and “i” in the group on the right.

  • The red squares now form a similar pattern in both Cayley tables.

  • Let’s move on to the next element in the group of integers mod 4, the number 1.

  • Well go through and color all squares containing a 1 green.

  • We do the same thing to the group on the right. The next uncolored element is “i”, so

  • let’s highlight all squares with an “i” green.

  • Notice the pattern of green in both tables is the same.

  • Let’s keep moving. In the integers mod 4, highlight all the 2’s

  • blue. And in the next group, highlight the “-1s”

  • blue. Once more, the blue pattern on the left is

  • the same as the blue pattern on the right. And well fill in the squares with the last

  • number using purple. Same pattern, again.

  • By coloring the squares a different color for each element, we can clearly see the patterns

  • for both Cayley tables are exactly the same. Theyre identical groups!! They just use

  • different elements and a different operation, but other than that, the groups are equivalent.

  • In both groups, if you combine a green element with a blue element, you get a purple element...

  • Similarly, a blue combined with a blue gives you a red, and so on.. Any such statement

  • about one table applies to the other. We say these two groups areisomorphic”, which

  • meansequal form.”

  • Now that weve seen a couple of examples, let’s talk about things more generally.

  • Suppose we have two groups G and H. They can be ANY kinds of groups: finite, infinite,

  • commutative, non-commutative, you name it. Now pick any 2 elements X and Y in the group

  • G. If we combine X and Y, we get a third element

  • in G. Well pronounce this “X times Y” even

  • though the operation can be something completely different than multiplication.

  • For the groups G and H to have similar group behavior, X, Y, and “X times Y” in G must

  • correspond to elements in the group H.

  • The mathematical way to write down this correspondence

  • is with a function. So we want a function F from G to H that sends

  • this part of the multiplication table for G to a similar part of the multiplication

  • table for H. F sends X to “F of X”, “Y” to “F

  • of Y”, and “X times Y” to “F of X times Y”

  • Here’s the critical observation. In the table for H, this square must also

  • contain “F of X” times “F of Y” In other words, “F of X” times “F of

  • Y” must equal “F of X times Y”. This type of function is the mathematical

  • tool we use to compare two groups.

  • We now have a tool for comparing two groups G and H. It is any function F from a group

  • G to a group H such that “F of X times Y” equals “F of X” times “F of Y.” We

  • call F a “group homomorphism”, or just a “homomorphismfor short. One thing

  • to be careful about is the operation on the left is the group operation in G, while on

  • the right it’s the group operation in H.

  • As an example, define the function F from the integers Z to itself by “F of X equals

  • 2X.” In this example, the group operation is regular

  • addition. To check if it’s a homomorphism, we only

  • have to show that “F of X + Y” equals “F of X” plus “F of Y”

  • From the definition of F, “F of X+Y” equals 2 times “X+Y”.

  • And on the right-hand-side, “F of X” equals “2X”, and “F of Y” equals “2Y”.

  • “2 times X+Y” DOES equal 2X plus 2Y, so F is a homomorphism.

  • Notice that the output of F is the set of even integers.

  • The even integers are a subgroup of the integers. Well explore this and other properties

  • of homomorphisms later.

  • After Groups, the idea of a Homomorphism is quite possibly the most important concept

  • in Abstract Algebra. It allows us to connect similar groups and identical groups. Homomorphisms

  • are also an essential tool for identifying the fundamental building blocks of groups.

  • And as we move forward through abstract algebra, this idea will recur again and again. We will

  • introduce a new object - rings, fields, vector spaces - and then define homomorphisms between

  • them.

  • The set of people who support us on Patreon is a finite but important group. They are

  • homomorphic to the Magic Group. This famous group is something I just made up as a way

  • to encourage you to support us financially on Patreon…{magic!}

Imagine, if you will, that we have two groups which we will creatively callGroup 1”

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B1 中級

群同型 - 抽象代数 (Group Homomorphisms - Abstract Algebra)

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    林宜悉 に公開 2021 年 01 月 14 日
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