Show that there are y,z such that y,z commute and their order is m and n

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Consider $x^s$ and $x^t$. What can you say about their orders and how can you use them to construct $y$ and $z$? In summary, the conversation is about an exercise where we need to show that if an element $x$ in group $G$ has order $mn$ with $(m,n)=1$, then there exist two elements $y$ and $z$ where $x=yz$ and $y$ and $z$ commute with each other, and have order $m$ and $n$ respectively. The hint given is to consider powers of $x$ and use the fact that $(m,n)=1$ to find suitable integers for constructing $y$ and
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mathmari
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Hey! :eek:

I got stuck at the following exercise:

If $x \in G$ has order $mn$ with $ (m,n)=1 $, show that there are $y,z$ with $ x=yz $ such that $y$,$z$ commute and they have order $m$ and $n$ respectively.

Could you give me some hints?? (Wondering)
 
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mathmari said:
Hey! :eek:

I got stuck at the following exercise:

If $x \in G$ has order $mn$ with $ (m,n)=1 $, show that there are $y,z$ with $ x=yz $ such that $y$,$z$ commute and they have order $m$ and $n$ respectively.

Could you give me some hints?? (Wondering)
Hint: think about powers of $x$.
 
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Another hint: Since $(m,n) = 1$, there are integers $s$ and $t$ such that $1 = sm + tn$.
 

What does it mean for two elements to commute?

Two elements commute if their order of operations does not affect the final result. In other words, if you switch the order of the two elements, the end result remains the same.

Why is it important to show that two elements commute?

Showing that two elements commute is important in many areas of mathematics, particularly in group theory and linear algebra. It allows for simpler calculations and proofs, and can reveal important properties and relationships between the elements.

How do you prove that there are two elements that commute?

To prove that there are two elements that commute, you must show that there exists some order in which the two elements can be multiplied and still result in the same value. This can be done through direct calculation or by using known properties and theorems.

What is the significance of proving that the order of the commuting elements is m and n?

The order of the commuting elements, m and n, represents the number of times the elements must be multiplied by themselves to result in the identity element. This can provide important information about the structure and properties of the elements and their relationship to each other.

Can you give an example of two elements that commute with orders m and n?

One example of two elements that commute with orders m and n is y = 2 and z = 3 in the group of integers modulo 6. In this case, m = 2 and n = 3, and we can see that yz = 2*3 = 6 = 0 (mod 6) and zy = 3*2 = 6 = 0 (mod 6), showing that these two elements commute.

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