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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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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 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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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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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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Cayley Table of Elementary Abelian Group $E_8$

I read about elementary abelian group $E_8$ at https://groupprops.subwiki.org/wiki/Elementary_abelian_group:E8#Definition. I've performed some searches on other sites and have yet to come across a Cayley table for it. Any leads on where to find one?
bblohowiak
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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 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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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…
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When does a multiplication table form a category?

Fix a (finite) set $A$. Say you are given a Cayley (multiplication) table for $A$: an $|A| \times |A|$ matrix, where each row and column corresponds to exactly one element of $A$, and the entries of the matrix are elements of $A \cup \{ - \}$. For…
Jordan Mitchell Barrett
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Understanding cyclic groups with Cayley tables

Question. Let $G = \{a,b,c,d,f\}$. Given that $(G, \cdot)$ is a cyclic group with $G=\langle d \rangle$ and Cayley table: \begin{array}{c|cc} \cdot & a & b & c & d & f\\ \hline a& c & a & f & b & d \\ b& a & b & c & d &f \\ c& f& c& d& a& b…
Charles Grealy
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Completing the Cayley table given certain information

Question. Let $G = \{1,2,3,4\}$. Given that $(G, \cdot)$ is a group with identity $3$ and that $o(x) = 2$ for each $x \in G \setminus \{3\}$, complete the Cayley table. I'm trying to break apart each statement in hopes to understand how I should…
Charles Grealy
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