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Questions tagged [cayley-table]

For questions about Cayley tables, a table that describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table.

A Cayley table, after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table. Many properties of a group — such as whether or not it is abelian, which elements are inverses of which elements, and the size and contents of the group's center — can be discovered from its Cayley table.

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Is there an easy way to see associativity or non-associativity from an operation's table?

Most properties of a single binary operation can be easily read of from the operation's table. For example, given $$\begin{array}{c|ccccc} \cdot & a & b & c & d & e\\\hline a & e & d & b & a & c\\ b & d & c & e & b & a\\ c & b & e & a &…
celtschk
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Can nonisomorphic groups have near-identical Cayley tables?

Nonisomorphic groups can have very similar multiplication (Cayley) tables. For example, the two…
Edward H
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Cayley Table Sudoku

Given an $n\times n$ grid partially filled with the numbers $0,\ldots, n-1$, we can play a Sudoku-like game by trying to fill in the rest of the grid so that the end result is the Cayley table for a group (edit: where position $i,j$ in the table…
volcanrb
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Cayley table group visualization

how can I make graphics like this? random colors. I got a script in GAP that prints rows of numbers but I want it colored just random colors G:=Units(Integers mod 2^3); n:=Order(G); M:=MultiplicationTable(G); for i in [1..n] do for j in [1..n]…
ryu jin
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In a Cayley table, which Group axioms fail when an entry appears twice in a row or a column?

In a Cayley table, which Group axioms fail when an entry appears twice in a row or a column? It's obviously not the Closure axiom, and after some inspection, I believe the Inverses axiom does fail. However, I'm not so sure how to show whether or…
Stephen
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Complexity of testing if a binary operation is a group

Given a binary operation specified as an $n \times n$ Cayley table, what is the complexity of the best deterministic algorithm for testing if the binary operation is a group? There's a fairly simple deterministic $O(n^2 \log n)$ algorithm which…
Qudit
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Closure under matrix multiplication for 2x2 matrices

I need to show that this set is closed under matrix multiplication, is there a better way than doing it via a Cayley table? Or rather I assume there is and I just can't get my head around it. Any help would be greatly…
ifearthenight
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Cayley table property of an infinite group

A Cayley table of an finite group has to have every element exactly once in every row and exactly once in every column. Proof that every element of a group has to be at most once in every row and at most once in every column: Let $(G, \circ)$ be a…
Martin Thoma
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Will this determinant of the matrix of determinants of the transformations of the group applied to a square matrix always be zero?

Background Let $A = \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right]$ be a matrix in $\mathbb{R}^{2 \times 2}$. While matrices are often used to represent a variety of linear transformations, including rotations, here I am transforming the…
Galen
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Cayley Table Wikipedia Notation

Wikipedia says that this is a Cayley table of Dih4: Can somebody explain what are these numbers/colors mean? Wikipedia page on Cayley table uses letters and does not mention any numbers. Also what are these small tables with red squares on top/left…
IgorStack
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Find a Group Given 2 Rows of a Cayley Table

I am working through Maxfield and Maxfield's Abstract Algebra and Solution by Radicals, and the following exercise is included after discussing permutations and Cayley's Theorem: Two rows of a certain Cayley table are r q s 1 u v t w q s 1 u v t w…
RHyp
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How to create multiplication table ("Cayley table") for an algebra or class of algebras?

I am studying universal algebra and getting familiar with the concept of variety of algebra. As far as I understand, a variety is just a class of all algebras satisfying given set of identities. Also, a variety is always closed under homomorphic…
Tereza Tizkova
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Does having exactly one of each element in every row and column of a Cayley table ensure that it’s a group?

A group must have a Cayley table in which each row and column has one and only one of each element. This can be proved by considering the opposite: suppose one row of a set’s Cayley table did not contain a particular element. That is, let AX not…
Thomas Delaney
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Cayley table with the identity along a diagonal

I have a Cayley table with four elements and a binary structure $*$. I know that if I have the same element along the main diagonal (from top right corner to bottom left corner), then the set is Abelian. What can I say about the set if the table…
user23793
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Is there an algorithm to check whether given subgroup contained inside the Frattini subgroup?

I am new to algorithmic group theory. I have the following question: Let $G$ be a group. The Frattini subgroup of $G$ is the intersection of all maximal subgroup of $G$, denoted by $\Phi(G)$. It is also equal to the set of all non-generating…
Pranjal
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