# 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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### 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 ...
1 vote
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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 ...
1 vote
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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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### 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. ...
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### Latin Square Problem: Indempotent Commutative Quasigroup of Order 7 [closed]

I missed the lecture that my professor went over Latin Squares and Idempotent Commutative Quasigroups. I understand it's essentially like the puzzle game Sudoku. I realize there are multiplication ...
1 vote
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### Make an addition and multiplication table for ring $\Bbb{Z}_{12}$ with ideal $\left \{ 0,3,6,9 \right \}$.

Make an addition and multiplication table for ring $\Bbb{Z}_{12}$ with ideal $\left \{ 0,3,6,9 \right \}$. I know how to make addition/multiplication tables, but I am confused as to how to find the ... 1k views

### 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&...
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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 ...
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### Up to what level can associativity be guaranteed?

My question is generated from the following question: It turns out that the inverse of product with an assumption of inverse existence is a necessary condition of associative. Then is there any set ...
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### Given a multiplication table for a set $G={a,b,c,d}$, determine whether it is a group.

Given the following multiplication table: How to determine whether it is a group? I know that $a$ must be the identity element, since for all $x\in G$, $a \circ x = x \circ a = x$. However, I cannot ... 1 vote
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### heuristics for determing if cayley tales are isomorphic

I've recently plunged into understanding the basics of group theory, mostly out of sheer fascination, and all sorts of interesting questions are coming to mind. Please remember I'm new at this; if ...
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 ...