If G is a group of odd order and x ∈G, show that x^-1 is not in cl(x).

Name: Problem 11, Chapter 24: Contemporary Abstract Algebra
Uploaded: 2020-02-04T00:04:19-08:00
Duration: 10 min 24 s
Channel: Heather Zimmers
Description: If G is a group of odd order and x ∈G, show that x^-1 is not in cl(x).

step 1 Okay, this is problem number 69, and it's a proof. We're asked to prove that G of X is even, if

If G is a group of odd order and x ∈G, show that x^-1 is not...

Key Concepts

Element Order and 2-Torsion

The order of an element in a group is the smallest positive integer such that raising the element to that power yields the identity element. In groups of odd order, elements cannot have order 2 because 2 would not divide the total number of elements. This absence of 2-torsion elements means that no non-identity element can be equal to its own inverse, a fact which is crucial for proving that an element’s inverse does not lie in its conjugacy class under nontrivial conditions.

Odd Order Groups

A group of odd order is one where the total number of elements is odd. This fact implies that no element other than the identity can have order 2 since 2 does not divide an odd number. This property is key in many arguments in group theory because it rules out the possibility of nontrivial elements being their own inverses, thus influencing the structure of conjugacy classes in such groups.

Conjugacy Classes

A conjugacy class in a group is the set of elements obtained by conjugating a given element by all elements in the group. This concept partitions the group into equivalence classes under the relation of conjugation. The properties of conjugacy classes are central in understanding the symmetry and internal structure of groups, particularly in studies involving class equations and representations.

Conjugation

Conjugation is the operation defined by g?¹xg for elements g and x in a group. This operation relates elements that are similar in structure and shows how group elements transform into one another. In the context of the problem, conjugation is used to investigate whether an element and its inverse can belong to the same conjugacy class, which has implications for the symmetric structure of the group.

Inverse Elements

In group theory, every element has a unique inverse such that the product of the element and its inverse is the identity element. Understanding the properties and behavior of inverse elements is critical for analyzing group structure and dynamics. The fact that an element’s inverse cannot coincide with its conjugate (other than in trivial cases) in an odd order group relies on the uniqueness of inverses and restrictions on element orders.

Transcript

00:04 Okay, this is problem number 69, and it's a proof.

00:10 We're asked to prove that g of x is even, if and only if f of x is even, and also g of x is odd, if and only if f of x is odd, where g of x equals 1 over f of x.

00:25 Another way to think of that is g of x and f of x are reciprocal functions.

00:31 So to do this proof thoroughly and correctly, it's really quite a long process.

00:38 I'm not sure if your teacher or instructor is expecting you to do a purely theoretical mathematical proof, but i'm going to do it justice here and work through a pure mathematician's proof.

00:52 So in order to prove an if and only if statement, which is called a biconditional statement, which means it's going in both directions, we need to prove two things.

01:02 So i've broken it into part one and part 2.

01:06 We need to prove that if f of x is even, then g of x is even.

01:11 And we also need to prove that if g of x is even, then f of x is even.

01:16 And if we can prove both of those things, then we can conclude that g of x is even if and only if f of x is even.

01:25 Okay, so here we go through part one of the first of two proofs.

01:30 So buckle up.

01:32 This is going to be quite a ride.

01:34 So if f of x is even then g of x is even.

01:38 So our given is that f of x is even.

01:45 What does it mean for a function to be even? well, it means that opposite x values will have the same y value.

01:52 So we can say then f of x equals f of the opposite of x.

02:01 That's a way to algebraically state that a function is even.

02:06 Now we know that g of x is 1 over f of x.

02:15 And according to the statement above if f of x is equal to f of the opposite of x i could do a substitution and show that g of x is equal to 1 over f of the opposite of x well we also know that g of the opposite of x just by definition is 1 over f of the opposite of x if you just take the function g of x equals 1 over f of x and put the opposite of x in both places.

02:54 Okay, well look at what we have here.

02:56 We have that 1 over f of the opposite of x is equal to g of x, and we also have that 1 over f of the opposite of x is equal to g of the opposite of x.

03:11 So we can substitute and say that g of x is equal to g of the opposite of x.

03:20 And that tells us that g of x is even.

03:27 So we've just proven that if f of x is even, then g of x is even.

03:34 So that part is done.

03:37 Now we go the other direction.

03:38 Part two, prove if g of x is even, then f of x is even.

03:43 So this time our given is that g of x is even.

03:49 What does that mean? an even function, just like before, opposite x coordinates have equal, y coordinates.

03:59 So we can say then g of x equals g of the opposite of x.

04:10 Now what is g of x? g of x is 1 over f of x.

04:21 Well if g of x is equal to g of the opposite of x, then 1 over f of x is also equal to g of the opposite of x.

04:32 But what is g of the opposite of x? by definition we know that g of the opposite of x is 1 over f of the opposite of x just by taking the relationship we had in the beginning and plugging the opposite of x in to both inputs okay once again we can do a substitution we see that g of the opposite of x is equal to 1 over f of x and it's also equal to 1 over f of the opposite of x so that tells us that 1 over f of x of that x is equal to 1 over f of the opposite of x...

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