Abstract Algebra: Proving/Disproving |a|=|b| if |a^2|=|b^2|

In summary, the statement "If |a^2|=|b^2|, prove or disprove that |a|=|b|" can be disproved by providing a counterexample, such as in the group Z20 with a = 2 and b = 4, where |a|=10 and |b|=5, but |a^2|=5 and |b^2|=5. However, this may not be true for all groups and a more general solution is needed.
  • #1
tyrannosaurus
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Homework Statement



If |a^2|=|b^2|, prove or disprove that |a|=|b|.

Homework Equations


The hint I was given is that let a be an element of order 4n+2 and let the order of b=a2


The Attempt at a Solution


I can disprove this by looking at examples, such as in the group Z20 with letting a =2 and b=4, the |a|=10 and |b|=5, but |a^2|=5 and |b^2|=5. But I know that this does not disprove it for all groups, I need a more general solution. If anyone can help me on this it would be great.
 
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  • #2
I'm going to go on a limb and say the statement is true for some groups (for example in Z3). When they say to disprove a conjecture, all that means is find a counterexample.
 

FAQ: Abstract Algebra: Proving/Disproving |a|=|b| if |a^2|=|b^2|

What is abstract algebra?

Abstract algebra is a branch of mathematics that studies algebraic structures such as groups, rings, and fields. It deals with generalizations of arithmetic operations, such as addition and multiplication, and studies their properties and relationships.

What does it mean to prove/disprove |a|=|b| if |a^2|=|b^2|?

This statement means that if the absolute value of the square of two numbers, a and b, are equal, then the absolute value of the numbers themselves must also be equal. In other words, if the squares of two numbers have the same magnitude, then the numbers themselves have the same magnitude.

How can this statement be proven or disproven?

This statement can be proven by using the properties and theorems of abstract algebra, specifically those related to groups and their operations. By manipulating the equation and using logical reasoning, we can show that the statement holds true for all possible values of a and b. On the other hand, it can be disproven by providing a counterexample, i.e. two numbers whose squares have the same magnitude but the numbers themselves have different magnitudes.

Why is this statement important in abstract algebra?

This statement is important because it highlights the concept of isomorphism, which is a fundamental concept in abstract algebra. Isomorphism refers to the idea that two algebraic structures may appear different, but they are essentially the same in terms of their properties and relationships. In this case, the statement shows that the group of positive real numbers under multiplication is isomorphic to the group of positive real numbers under addition.

Can this statement be applied to other algebraic structures?

Yes, this statement can be applied to other algebraic structures, such as rings and fields, as long as they have the necessary properties and operations. However, the statement may need to be modified to fit the specific structure. For example, in a ring, the statement would be |a|=|b| if |a^2|=|b^2| and |a+b|=|b+a|, since addition is also a defined operation in rings.

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