# 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.

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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 ...
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### Ordering of group elements in MultiplicationTable vs CosetTable in GAP

I construct a finitely presented discrete group $G$ in GAP, a normal subgroup $H\triangleleft G$ of finite index $N$, and the factor group $G/H$ of order $N$. Assume for simplicity that $G$ has two ...
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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 ...
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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 groups \begin{align*} \mathbb{Z}/9\mathbb{Z}&=\{\overset{a}{0},\overset{b}{1},\overset{c}{2},\...
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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 ...
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### Determining the binary operation from a Cayley Table

I have tried a lot of things with this Cayley Table (several Julia/Python scripts which iterate over various functions, symbolic regression, semi-manually trying various permutation groups, octonions, ...
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### Properties of Groups Seen in Cayley Tables

The following is a simple exercise from "Guide to Abstract Algebra 1st Ed Carol Whitehead". Problem 6.2b.1 The following is part of a Cayley table for a group $G = {a,b,c,d}$, with respect ...
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### Composition of two cube rotation symmetries, that lie along perpendicular axis.

Want to find the orbits of the group of $24$ rotations of the cube. The rotations about the axis, formed by the midpoints of the anti-podal faces parallel to the $x-z$ plane have the $3$ non-trivial ...
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### What is the most efficient way of storing group information? How does GAP do it?

What is the most efficient way of storing the information contained in group's Cayley table on a disk? Cayley tables grow with $|G|^2$, and for some orders ($p^k$ for growing $k$) there are just too ...
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### How to construct group table without the hint of subgroup $H =\{(1), (12), (34), (12)(34)\}$ is a subgroup in $S_4$

Given that $G=\{e, u,v,w\}$ is a group of order $4$ with $u^2=v, v^2=e$. Construct its multiplication table. Does such a group exist? The table has seven unfilled entries, that have no way to be ...
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### Terse filling of the Cayley table

Given a group of order $n$, what are the least number of elements to specify on the Cayley table to specify a group? Example Consider this $Z_4$ group table for example: Due to abelian-ness I can ...
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### I cannot fill the Cayley table for the group of quaternion units. How to calculate the value of $a\theta$? (Herstein "Topics in Algebra 2nd Edition")

I am reading "Topics in Algebra 2nd Edition" by I. N. Herstein. The following problem is Problem 21 on p.81 in this book: Let $G$ be the group $\{e,\theta,a,b,c,\theta a,\theta b,\theta c\}$...
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### I want to prove that every group of order $4$ is going to be isomorphic to these two.

I am trying to show that there exist only $2$ non-isomorphic groups of order $4$. I found the groups using Cayley Tables, (I think one is called the Klein group that I found, and the other one is a ...
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### When we classify groups of order $n$, can I skip to check if the associativity law holds after I complete Cayley Table?

For example, classify groups of order $4$. Let $G=\{a,b,c,d\}$ be a group whose order is $4$. $G$ must have an identity element $e$. Without loss of generality, we can assume $d$ is the identity ...
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### What is the formal definition of a Cayley table?

What is the formal definition of a Cayley table? I am not interested merely in Cayley tables for groups, I am interested in general Cayley tables for non-empty finite magmas. Also, another question is,...
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### Does this set along with $\text{mod } n$ form group?

Does $\left\{0, 1, 2\right\}$ along with the operation of addition $\text{mod } 6$ form a group? I have many practice questions like this, and I know I have to check closure, associativity, identity, ...
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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 ...
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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, ...
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### Cayley tables and associativity [duplicate]

I don't see how can one check associative property in a Cayley table. For contrast, one can find identity element, inverses and commutativity. But how you check associativity? I understand that ...
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### Is there bijection between Cayley tables and (finite) groups (if the order of rows and colums doesn't matter)?

I have questions about structure of groups. Is there bijection between Cayley tables and (finite) groups? So, Cayley table is a table of permutations of finite elements in which none element repeat in ...
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### How do we construct cayley tables for fields without using Lagrange's Theorem?

I'm just learning about sets, groups and fields, and I'm not sure how I'd go about making the addition cayley table for a field with, say, 4 or 5 elements. I know that, for example, 0 plus any element ...
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### Matrix Group under multiplication

Suppose the collection $\{A_1, A_2,...,A_k\}$ forms a Group under matrix multiplication, where each $A_i$ is an $n \times n$ real matrix. Let $A = \sum_{i=1}^{k} A_i$ Show that $A^2 = kA$ If the ...
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### Does Cayley Table that is symmetrical along the diagonal with identities really mean the group is Abelian?

I am having this doubt because Cayley Table for integers under mod 4 seems to be Abelian but its Cayley Table doesn't seem to be symmetrical. https://www.youtube.com/watch?v=BwHspSCXFNM&list=...
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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 ...
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### Number of nonisomorphic structures are there to the possible binary structures on the set $\{a,b\}$?

My question stems from the following question: How many non-isomorphic binary structures on the set of $n$ elements? It goes on to say that for the $16$ possible binary structures on the set $\{a,b\}$...
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