Cayley’s theorem

Cayley’s theorem is one of the fundamental equations in graph theory. It says that any graph can be drawn without edge crossings with an odd number of edges, by alternately adding edges that start/end on existing edges and following their directions.” 

What is Cayley’s Theorem?

Cayley’s theorem is a statement about the structure of groups that was first proved by Arthur Cayley. It is often used in algebra, especially in contexts where linear transformations are involved. The theorem states that a group can always be represented as the product of its left and right cosets (which then turns out to be a simpler form than the original group). That is, if H = {a | ∀x ∈ G, xax} then there will exist another subgroup M = {x |aax} such that H ⋅ M = G

Cayley’s theorem can easily be connected with many other concepts and facts. For example, it is related to the cosets of a given subgroup in a group and the way to obtain the quotient groups from groups.

Theorem: A group G can be represented as H ⋅ M, where H is a normal subgroup and M = {x |ax}.

Cayley’s theorem about groups is perhaps one of the most powerful tools for dealing with all types of abstract structures that have an underlying algebraic structure. This is because as Arthur Cayley who discovered this theorem first proved, any group can be represented as a direct product of its left and right cosets.

Cayley–Hamilton theorem:

Cayley’s theorem can be stated in terms of matrices and linear algebra as Cayley–Hamilton theorem. The latter generalises the associative law of multiplication to the case when it is applied to a matrix times another matrix.

Formula of Cayley–Hamilton theorem: p(λ) = det(λI − σ)

Cayley–Hamilton theorem gives an associative law of matrix multiplication.

Example of Cayley-Hamilton Theorem:

When the matrix ℓ is represented by a row of zeros, the Cayley–Hamilton theorem states:

Cayley’s theorem could be used to show that every graph can be drawn without edge crossings. This is known as planar graphs. In fact, every graph can be decomposed into planes or areas without any crossings. So it can be said that Cayley’s theorem is a tool to prove any result on planar graphs. However, on a geometric plane (or Euclidean plane), the edges are not separated and therefore the graph has no edges. Thus one cannot draw this plot on such a surface.

What is a Cayley table?

A Cayley table is a way of representing the elements and the operation(s) that can be performed on them in a way that can be used to simplify complicated arithmetic, using addition and multiplication separately. The table shows all combinations of the elements, their inverses, and/or their product; usually, it also shows whether or not the elements have a multiplicative inverse. A Cayley table is composed of all the possible combinations of, say, the elements a, b, c and d with their inverses i and i−1. The entries in this table are called Cayley numbers. In mathematics(not only in graph theory), every graph can be described by its vertices (points), edges (lines) and faces (two adjacent vertices). But at the beginning of the last century there was a problem: how to represent these graphs? Janos Bolyai solved this problem by developing a new method for talking about graphs — linear algebra.

Today we describe a graph as: We know that each graph has its ‘vertices’, which are points that determine these graphs. The ‘edges’ are the lines that connect these points. And ‘faces’ are the two points that form a line. This is why every graph has edges and faces, which means it has a dual graph.

Significance of Cayley’s theorem:

Cayley’s theorem is about representation of graphs. It states that every graph can be represented as the product of its left and right cosets. Cayley’s theorem has many applications. One of them is solving problems. This method is well known in graph theory, which shows how a problem can be solved with the help of Cayley’s theorem. Cayley’s theorem has many applications in Mathematics, Chemistry and Physics. Some people say that it even exists in Social Sciences and linguistics, but this part is doubtful. It describes different types of groups, some relations etc. Cayley’s theorem is also often referred to as Euler’s theorem, which is an identity about the composition of functions. Finally, Cayley’s theorem has applications in quantum computing. Thanks to it it is possible to create qubits (quantum bits), which are the building blocks of quantum computers and use them in quantum algorithms.

Conclusion:

Cayley’s theorem is one of the most important and famous theorems in Mathematics. It could be applied to almost all fields, including Biology, Chemistry and Physics. He also discovered Cayley graphs, which are a way of representing groups using special diagrams. So, Cayley was a very talented person who worked in various fields of mathematics.