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.

Filter by
Sorted by
Tagged with
0 votes
2 answers
50 views

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 ...
user avatar
  • 3,435
2 votes
1 answer
64 views

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 ...
user avatar
0 votes
0 answers
42 views

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 $\{e,\theta,a,b,c,\theta a,\theta b,\theta c\}$...
user avatar
  • 6,117
2 votes
3 answers
138 views

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 ...
user avatar
  • 409
1 vote
1 answer
50 views

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 ...
user avatar
  • 6,117
0 votes
0 answers
27 views

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,...
user avatar
  • 13.3k
0 votes
1 answer
71 views

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, ...
user avatar
  • 409
5 votes
2 answers
84 views

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 ...
user avatar
  • 1,328
3 votes
2 answers
140 views

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, ...
user avatar
0 votes
0 answers
38 views

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 ...
user avatar
  • 1,192
3 votes
0 answers
55 views

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 ...
user avatar
  • 1,192
0 votes
1 answer
73 views

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 ...
user avatar
3 votes
2 answers
212 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 ...
user avatar
3 votes
4 answers
206 views

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 &...
user avatar
1 vote
1 answer
141 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....
user avatar
1 vote
1 answer
49 views

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 ...
user avatar
1 vote
0 answers
65 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 ...
user avatar
  • 365
1 vote
1 answer
109 views

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....
user avatar
  • 367
0 votes
1 answer
87 views

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 $x_{ij}$ ...
user avatar
  • 301
2 votes
1 answer
82 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 ...
user avatar
  • 341
1 vote
1 answer
201 views

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 & ...
user avatar
1 vote
1 answer
51 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 ...
user avatar
  • 604
-2 votes
2 answers
112 views

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, ...
user avatar
0 votes
1 answer
100 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 ...
user avatar
1 vote
0 answers
28 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 ...
user avatar
  • 161
0 votes
2 answers
276 views

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 ...
user avatar
  • 101
1 vote
4 answers
74 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 ...
user avatar
  • 3,435
-2 votes
2 answers
104 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 ...
user avatar
  • 3,435
1 vote
1 answer
94 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, ...
user avatar
0 votes
0 answers
94 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 ...
user avatar
  • 3,435
2 votes
2 answers
493 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 ...
user avatar
1 vote
1 answer
163 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 ...
user avatar
0 votes
2 answers
129 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 ...
user avatar
  • 365
4 votes
0 answers
89 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 ...
user avatar
2 votes
1 answer
43 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 "...
user avatar
  • 23
1 vote
3 answers
125 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 ...
user avatar
2 votes
2 answers
135 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 ...
user avatar
0 votes
0 answers
207 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=...
user avatar
  • 1,426
4 votes
1 answer
363 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 ...
user avatar
1 vote
2 answers
517 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 $\{a,b\}$...
user avatar
  • 5,102
9 votes
3 answers
1k 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. ...
user avatar
  • 3,522
0 votes
2 answers
381 views

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 ...
user avatar
  • 135
1 vote
1 answer
844 views

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 ...
user avatar
4 votes
1 answer
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&...
user avatar
4 votes
3 answers
768 views

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 ...
user avatar
0 votes
1 answer
126 views

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 ...
user avatar
0 votes
2 answers
661 views

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 ...
user avatar
1 vote
0 answers
51 views

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 ...
user avatar
2 votes
1 answer
463 views

How do you make a Cayley table for the molecule ethane from the D3h group?

Using the simulation website below, you can see the symmetry elements (axes and planes) of the molecule ethane. http://symmetry.otterbein.edu/gallery/index.html I am trying to construct a Cayley ...
user avatar
9 votes
1 answer
186 views

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 ...
user avatar
  • 3,160