An interesting question from “Group Theory: A First Journey,” (page 4, section 2.3).
I am currently studying the manuscript Group Theory: A First Journey by Vipul Naik. It is available from the web page. In this manuscript the author proposes the following question:
Suppose the ordering of the elements in the rows and columns is the same. Then what kind of multiplication table would a magma have if it were to be a group? More generally, what are the constraints on the multiplication table corresponding to each of the properties that we can talk of for a binary operation?
(page 4, section 2.3)
No matter from what angle I have thought about this question, I can't seem to grasp what the author wants from his readers.
Anyone?
group-theory binary-operations magma
edited Nov 28 at 22:51
Shaun
8,344113578
asked Feb 6 '14 at 14:50
Valera Rozuvan
304111
add a comment |
I am currently studying the manuscript Group Theory: A First Journey by Vipul Naik. It is available from the web page. In this manuscript the author proposes the following question:
Suppose the ordering of the elements in the rows and columns is the same. Then what kind of multiplication table would a magma have if it were to be a group? More generally, what are the constraints on the multiplication table corresponding to each of the properties that we can talk of for a binary operation?
(page 4, section 2.3)
No matter from what angle I have thought about this question, I can't seem to grasp what the author wants from his readers.
Anyone?
group-theory binary-operations magma
edited Nov 28 at 22:51
Shaun
8,344113578
asked Feb 6 '14 at 14:50
Valera Rozuvan
304111
For example (as I understand it), if you want your structure to be commutative, your table has to be symmetric, since you want $ab=ba$. If you want every element to be invertible, then you want "1"-s in certain places...
– Ludolila
Feb 6 '14 at 14:53
Well, the identity and the inverse have nice enough descriptions in this context. For the associativity, it is a bit more tricky, since this involves more than two elements.
– Tobias Kildetoft
Feb 6 '14 at 14:55
See Latin square.
– lhf
Feb 6 '14 at 15:20
add a comment |
I am currently studying the manuscript Group Theory: A First Journey by Vipul Naik. It is available from the web page. In this manuscript the author proposes the following question:
Suppose the ordering of the elements in the rows and columns is the same. Then what kind of multiplication table would a magma have if it were to be a group? More generally, what are the constraints on the multiplication table corresponding to each of the properties that we can talk of for a binary operation?
(page 4, section 2.3)
No matter from what angle I have thought about this question, I can't seem to grasp what the author wants from his readers.
Anyone?
group-theory binary-operations magma
edited Nov 28 at 22:51
Shaun
8,344113578
asked Feb 6 '14 at 14:50
Valera Rozuvan
304111
I am currently studying the manuscript Group Theory: A First Journey by Vipul Naik. It is available from the web page. In this manuscript the author proposes the following question:
Suppose the ordering of the elements in the rows and columns is the same. Then what kind of multiplication table would a magma have if it were to be a group? More generally, what are the constraints on the multiplication table corresponding to each of the properties that we can talk of for a binary operation?
(page 4, section 2.3)
No matter from what angle I have thought about this question, I can't seem to grasp what the author wants from his readers.
Anyone?
group-theory binary-operations magma
group-theory binary-operations magma
edited Nov 28 at 22:51
Shaun
8,344113578
asked Feb 6 '14 at 14:50
Valera Rozuvan
304111
edited Nov 28 at 22:51
Shaun
8,344113578
asked Feb 6 '14 at 14:50
Valera Rozuvan
304111
edited Nov 28 at 22:51
Shaun
8,344113578
edited Nov 28 at 22:51
Shaun
8,344113578
edited Nov 28 at 22:51
Shaun
8,344113578
8,344113578
asked Feb 6 '14 at 14:50
Valera Rozuvan
304111
asked Feb 6 '14 at 14:50
Valera Rozuvan
304111
asked Feb 6 '14 at 14:50
Valera Rozuvan
304111
304111
For example (as I understand it), if you want your structure to be commutative, your table has to be symmetric, since you want $ab=ba$. If you want every element to be invertible, then you want "1"-s in certain places...
– Ludolila
Feb 6 '14 at 14:53
Well, the identity and the inverse have nice enough descriptions in this context. For the associativity, it is a bit more tricky, since this involves more than two elements.
– Tobias Kildetoft
Feb 6 '14 at 14:55
See Latin square.
– lhf
Feb 6 '14 at 15:20
add a comment |
For example (as I understand it), if you want your structure to be commutative, your table has to be symmetric, since you want $ab=ba$. If you want every element to be invertible, then you want "1"-s in certain places...
– Ludolila
Feb 6 '14 at 14:53
Well, the identity and the inverse have nice enough descriptions in this context. For the associativity, it is a bit more tricky, since this involves more than two elements.
– Tobias Kildetoft
Feb 6 '14 at 14:55
See Latin square.
– lhf
Feb 6 '14 at 15:20
For example (as I understand it), if you want your structure to be commutative, your table has to be symmetric, since you want $ab=ba$. If you want every element to be invertible, then you want "1"-s in certain places...
– Ludolila
Feb 6 '14 at 14:53
For example (as I understand it), if you want your structure to be commutative, your table has to be symmetric, since you want $ab=ba$. If you want every element to be invertible, then you want "1"-s in certain places...
– Ludolila
Feb 6 '14 at 14:53
Well, the identity and the inverse have nice enough descriptions in this context. For the associativity, it is a bit more tricky, since this involves more than two elements.
– Tobias Kildetoft
Feb 6 '14 at 14:55
Well, the identity and the inverse have nice enough descriptions in this context. For the associativity, it is a bit more tricky, since this involves more than two elements.
– Tobias Kildetoft
Feb 6 '14 at 14:55
See Latin square.
– lhf
Feb 6 '14 at 15:20
See Latin square.
– lhf
Feb 6 '14 at 15:20
add a comment |
1 Answer
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Consider the existence of unique inverses. What does this do to the rows and columns of the multiplication table? For example, what if $astar b = c$ but then $astar d = c$ too?
You'll also need an identity element, and in particular this must be a two-sided identity, meaning $estar x = x star e = x$. Without loss of generality we can take $e$ to be the first element by rows and columns. How should the first row and the first column be related?
As far as associativity goes, I'm not sure there's a particularly good way of describing this in terms of the group multiplication table. It just has to work. (If you can think of a good way, please let me know!)
answered Feb 6 '14 at 17:06
Alexander Gruber♦
20k25102171
1
About testing associativity, see math.stackexchange.com/questions/511682/….
– lhf
Feb 7 '14 at 9:59
add a comment |
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Consider the existence of unique inverses. What does this do to the rows and columns of the multiplication table? For example, what if $astar b = c$ but then $astar d = c$ too?
You'll also need an identity element, and in particular this must be a two-sided identity, meaning $estar x = x star e = x$. Without loss of generality we can take $e$ to be the first element by rows and columns. How should the first row and the first column be related?
As far as associativity goes, I'm not sure there's a particularly good way of describing this in terms of the group multiplication table. It just has to work. (If you can think of a good way, please let me know!)
answered Feb 6 '14 at 17:06
Alexander Gruber♦
20k25102171
1
About testing associativity, see math.stackexchange.com/questions/511682/….
– lhf
Feb 7 '14 at 9:59
add a comment |
Consider the existence of unique inverses. What does this do to the rows and columns of the multiplication table? For example, what if $astar b = c$ but then $astar d = c$ too?
You'll also need an identity element, and in particular this must be a two-sided identity, meaning $estar x = x star e = x$. Without loss of generality we can take $e$ to be the first element by rows and columns. How should the first row and the first column be related?
As far as associativity goes, I'm not sure there's a particularly good way of describing this in terms of the group multiplication table. It just has to work. (If you can think of a good way, please let me know!)
answered Feb 6 '14 at 17:06
Alexander Gruber♦
20k25102171
1
About testing associativity, see math.stackexchange.com/questions/511682/….
– lhf
Feb 7 '14 at 9:59
add a comment |
Consider the existence of unique inverses. What does this do to the rows and columns of the multiplication table? For example, what if $astar b = c$ but then $astar d = c$ too?
You'll also need an identity element, and in particular this must be a two-sided identity, meaning $estar x = x star e = x$. Without loss of generality we can take $e$ to be the first element by rows and columns. How should the first row and the first column be related?
As far as associativity goes, I'm not sure there's a particularly good way of describing this in terms of the group multiplication table. It just has to work. (If you can think of a good way, please let me know!)
answered Feb 6 '14 at 17:06
Alexander Gruber♦
20k25102171
Consider the existence of unique inverses. What does this do to the rows and columns of the multiplication table? For example, what if $astar b = c$ but then $astar d = c$ too?
You'll also need an identity element, and in particular this must be a two-sided identity, meaning $estar x = x star e = x$. Without loss of generality we can take $e$ to be the first element by rows and columns. How should the first row and the first column be related?
As far as associativity goes, I'm not sure there's a particularly good way of describing this in terms of the group multiplication table. It just has to work. (If you can think of a good way, please let me know!)
answered Feb 6 '14 at 17:06
Alexander Gruber♦
20k25102171
answered Feb 6 '14 at 17:06
Alexander Gruber♦
20k25102171
answered Feb 6 '14 at 17:06
Alexander Gruber♦
20k25102171
answered Feb 6 '14 at 17:06
Alexander Gruber♦
20k25102171
20k25102171
1
About testing associativity, see math.stackexchange.com/questions/511682/….
– lhf
Feb 7 '14 at 9:59
add a comment |
1
About testing associativity, see math.stackexchange.com/questions/511682/….
– lhf
Feb 7 '14 at 9:59
1
1
About testing associativity, see math.stackexchange.com/questions/511682/….
– lhf
Feb 7 '14 at 9:59
About testing associativity, see math.stackexchange.com/questions/511682/….
– lhf
Feb 7 '14 at 9:59
add a comment |
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For example (as I understand it), if you want your structure to be commutative, your table has to be symmetric, since you want $ab=ba$. If you want every element to be invertible, then you want "1"-s in certain places...
– Ludolila
Feb 6 '14 at 14:53
Well, the identity and the inverse have nice enough descriptions in this context. For the associativity, it is a bit more tricky, since this involves more than two elements.
– Tobias Kildetoft
Feb 6 '14 at 14:55
See Latin square.
– lhf
Feb 6 '14 at 15:20