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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.

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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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1 answer
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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 ...
volcanrb's user avatar
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23 votes
1 answer
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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},\...
Edward H's user avatar
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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 ...
Raman's user avatar
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1 answer
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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 ...
Penelope's user avatar
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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 ...
jiten's user avatar
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0 votes
1 answer
107 views

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 ...
user35443's user avatar
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2 answers
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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 ...
Cathartic Encephalopathy's user avatar
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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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2 votes
3 answers
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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 ...
eddie's user avatar
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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, ...
eddie's user avatar
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3 answers
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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 ...
Galen's user avatar
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2 answers
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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, ...
Tereza Tizkova's user avatar
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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 ...
Elmer Cado's user avatar
3 votes
2 answers
618 views

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 ...
Rudranath Palit's user avatar
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4 answers
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Group problem with a diagram

Let the set $M=\{a,b,c,d\}$ and a binary operation $\star$ described in the diagram : $$\begin{array}{c|cccc} \star & a & b & c & d \\ \hline a & a & b & a & b \\ b &...
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1 vote
1 answer
704 views

Multiplication/Cayley tables for the Dihedral Groups

I am currently doing a group theory problem, which asks for the multiplication table of the dihedral group $D_4$. Having looked up the answer online, I do not understand how some of the elements arose....
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2 votes
1 answer
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An example of an algebraic loop which has different L and R inverses?

Can anyone point me toward a simple example of a non-associative algebraic loop (i.e. a quasigroup with an identity) for which at least one element has a left inverse which is not equal to its right ...
KesterKester's user avatar
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149 views

Cayley table for the additive and mutiplicative operations in ($\Bbb F_7$, +, *)

I am to construct the Cayley tables for the additive and multiplicative operation in $(\Bbb F_7, +, *)$. I have started by stating The order (nr of elements) of a finite field must be a prime or ...
Mampenda's user avatar
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1 answer
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Fill in Cayley table for a commutative ring with unity

Let M={0,1,a,b} be a commutative ring with unity. I'm supposedt to fill in the Cayley tables. Can someone help me with the table for the multiply table. I think that $a*b=b*a=a$ do to commutative ring....
Erika's user avatar
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1 answer
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How to define a finite set S which is a non-abelian group under binary operation without commutativities except the trivial ones (see Cayley table)?

By "without trivial commutativities" I mean "$bc\neq cb$ for any $b,c \in S \setminus \{a\}$, where $a$ is the identity and $b\neq c$. I found out how to calculate the element $x_{ij}$ ...
dawid's user avatar
  • 335
2 votes
1 answer
237 views

Fill in a partly filled in table such that it makes the magma $(M,*)$ associative, commutative, has an identity element and has no zero-elements.

Below is a partly filled in table for a binary operation ($*$) on the set $M=\{a,b,c,d\}$. I am trying to fill in the rest such that the magma $(M,*)$ becomes associative, commutative, has an identity ...
NoName123's user avatar
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2 votes
1 answer
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Constructing the $Z_2 \times Z_2$ group table

In A. Zee's group theory book p. 47-49, he constructs the group table with four elements $\{I,A,B,C\}$ $\begin{array}{c|cccc} & I & A & B & C \\ \hline I & I & A & B & ...
mathemania's user avatar
1 vote
1 answer
62 views

Prove the group defined by the following relations has eight elements and is not isomorphic to $\Delta_4$

I'm working out the exercises in MacLane and Birkhoff's Algebra. The exercise in question is the II.5.8: where $\Delta_4$ stands for the fourth dihedral group. Now I'm stuck trying to answer the ...
Albert's user avatar
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-2 votes
2 answers
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What would a Cayley table of inverse semigroup look like?

I tried to construct a Cayley table of an algebraic structure called inverse semigroup. No success so far. I just end up with more complicated structure (monoid, group). Thank you kindly. I may think, ...
Josef Hlava's user avatar
0 votes
1 answer
127 views

Associativity indeed imply closure of binary operations... or what is wrong?

Similar questions have been asked without great success to answer. I read what I could about the problem, but no idea. I wrote to university professor, no response. There has been question about ...
Josef Hlava's user avatar
1 vote
0 answers
120 views

Using an operations table for the group $D_{6}$

With $H = \{ \rho_{0} , \rho_{3} \}$, how would I compute $( \mu_{1} , H)(\mu_{2} , H)$ and $( \mu_{2} , H)(\mu_{1} , H)$? I'm assuming I'd calculate $( \mu_{1} , \rho_{0} )( \mu_{2} , \rho_{3} )$ and ...
Matt L.'s user avatar
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2 answers
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Cayley Table of $(\mathbb{Z}_5^*, \cdot)$

1) Determine the Cayley Table of $(Z_5^*, \cdot)$ 2) determine which additive group has the exact same table. 3) Further determine an isomorphism between those two groups and prove by means of ...
M-xyz1's user avatar
  • 101
1 vote
4 answers
104 views

Unable to use property of order $2$ elements to prove that groups with diagonal elements $=e$ are abelian.

If $G$ is a group such that $x^2 = e$ for all elements $x$ in $G$, then show that $G$ is abelian. I wanted to propose a method, that is based on properties of order $2$ elements, that I observed is ...
jiten's user avatar
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-2 votes
2 answers
108 views

Request help in doubts of group table of groups of order $4$.

It is shown in section $1.3$ of ug level book on Group Theory by Louis W. Shapiro, a way to build up the possible multiplication tables for groups of order $4$. I know three properties only, as ...
jiten's user avatar
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1 vote
1 answer
551 views

Finding a subgroup of $S_5$ that is isomorphic to $D_5$

To find a subgroup, I calculated the entire Cayley Table for $D_5$ and looked for wherever there was an Identity. Then, I calculated my subgroup of $S_5$ accordingly. I know that $D_5$ = {$I, R_1, ...
flutterbug98's user avatar
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0 answers
116 views

Unable to understand how six substitutions (functions) form a group.

Have a doubt in exercise that is repeated in two texts (as stated at the end)- Suppose there are six elements $1,a,b,c,d,e$ whose laws of combination are given by the table. As per the table, there ...
jiten's user avatar
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2 votes
2 answers
904 views

Confusion about creating Cayley table for ($\mathbb{Z}_{18}^*,\times)$

On a test I ran into a question about a Cayley table. The question was "Given the group $(\mathbb{Z}_{18}^*,\times)$, construct the Cayley table." It also said that this group is sometimes referred to ...
Wouterds's user avatar
1 vote
1 answer
408 views

What is the identity element of 4

If $*$ is a binary operation taking the greater of two distinct numbers, construct a table for the operation on the set $S=\{1,2,3,4,5\}$. What is the identity element of 4? Is the operation ...
Sefiuoyedeji's user avatar
0 votes
2 answers
278 views

Cayley table for $D_4/\langle r^2\rangle$.

Write out the Cayley table for $D_4/\langle r^2\rangle$. I know that $D_4$ is the dihedral group with 4 sides (square) and $\langle r^2\rangle$ is the algebra generated by $r^2$ (a.k.a. taking the ...
yagayeet's user avatar
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4 votes
0 answers
125 views

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 ...
Jordan Barrett's user avatar
2 votes
1 answer
48 views

group isomorphisms not decomposable into relabelling or row-col exchange sequences

I am interested in the way groups represent abstract structure, but unfortunately I'm dyslexic in understanding the notation. As a novice in this area I see statements that group isomorphisms "...
Addy's user avatar
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1 vote
3 answers
260 views

Showing Associativity and Commutativity of a binary operation given by a Cayley table

Let $*$ be a binary operation on the set $S:=\{0,1\}$ given by the following Cayley table: \begin{array}{c|cc} * & 0 & 1\\\hline 0 & 0 & 1\\ 1 & 1 & 0 \end{array} If I wish to ...
Taylor Rendon's user avatar
2 votes
2 answers
206 views

Elements of $E^{\times},\cdot$ of the quotient ring $E:= \frac{\mathbb{Z}_3[X]}{\langle x^2 + x + 2\rangle}$

Consider the field $E:= \frac{\mathbb{Z}_3[X]}{\langle x^2 + x + 2\rangle}$. If I'm right the elements of the quotient ring can be found as: $$a_0 + a_1x + \langle x^2 + x + 2\rangle.$$ So we got the ...
Belgium_Physics's user avatar
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0 answers
460 views

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=...
user3000482's user avatar
  • 1,526
5 votes
1 answer
676 views

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 ...
bblohowiak's user avatar
1 vote
2 answers
807 views

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 $\{a,b\}$...
WaveX's user avatar
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10 votes
3 answers
2k views

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. ...
Stephen's user avatar
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