Finite Group Inverses: Proving $N_{ABC}=N_{CBA}$

In summary, a finite group inverse is an element in a finite group that, when multiplied by another element, produces the identity element. To prove that $N_{ABC}=N_{CBA}$ for finite groups, you can use the properties of group inverses and the associative property of group multiplication. This proof shows that the order in which elements are multiplied in a finite group does not matter and has applications in physics and chemistry. It can be applied to all finite groups as long as they follow the necessary properties.
  • #1
Fallen Angel
202
0
Hi,

I bring a new algebraic challenge ;)

Let $G$ be a finite group and $U,V,W\subset G$ arbitrary subsets of $G$.
We will denote $N_{UVW}$ the number of triples $(x,y,z)\in U\times V \times W$ such that $xyz$ is the unity of $G$, say $e$.
Now suppose we have three pairwise disjoint sets $A,B,C$ such that $G=A\cup B \cup C$

Prove that $N_{ABC}=N_{CBA}$.
 
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  • #2
A hint:

Start proving that for arbitrary $U,V\subset G$
$N_{UVG}=|U||V|$
and for arbitraty $U,V,W\subset G$
$N_{UVW}=N_{WUV}=N_{VWU}$
 

FAQ: Finite Group Inverses: Proving $N_{ABC}=N_{CBA}$

What is a finite group inverse?

A finite group inverse is an element in a finite group that can be multiplied by another element to produce the identity element. It is essentially the "opposite" of an element in a group, and when multiplied together, the result is the identity element (usually denoted as "e").

How do you prove that $N_{ABC}=N_{CBA}$ for finite groups?

To prove that $N_{ABC}=N_{CBA}$ for finite groups, you can use the properties of group inverses and the associative property of group multiplication. This means showing that $N_{ABC}$ and $N_{CBA}$ are both equal to the identity element when multiplied with each other.

What is the significance of proving $N_{ABC}=N_{CBA}$ in finite groups?

This proof shows that the order in which elements are multiplied in a finite group does not matter. It also demonstrates the commutative property of group multiplication, which states that the order of elements can be switched without affecting the outcome.

Can this proof be applied to all finite groups?

Yes, this proof can be applied to all finite groups. As long as the group follows the properties of group inverses and the associative property of multiplication, the proof will hold.

Are there any real-world applications of this proof in science?

Yes, this proof has applications in many areas of science, including physics and chemistry. In physics, it can be used to understand the commutative property of physical quantities and in chemistry, it can be used to analyze the symmetry of molecules and their reactions.

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