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About Symmetry About a Line

Date: 06/06/2012 at 06:48:57
From: diegs
Subject: Testing for commutativity

I know that if the table of an operation is symmetric about a diagonal
line from upper left to lower right, then the operation is commutative.

But in the case of the table below, I see symmetry from lower left to
upper right:

   *  a   b   c
   a |a | b | c|
   b |b | c | a|
   c |c | a | b|

Can I state this as one of the properties of a table which ensures that
the given operation is commutative?

Date: 06/06/2012 at 09:29:27
From: Doctor Peterson
Subject: Re: Testing for commutativity

Hi, Diegs.

You're right that a commutative operation corresponds to a table that is
symmetric about its diagonal.

And this table DOES have that symmetry; the diagonal you're talking about
goes from upper left to bottom right. So the symmetry (reflection) takes
cells from upper right to bottom left, across that line, just as you say
you see.

Were you looking at the wrong diagonal, or misunderstanding what symmetry
ABOUT a line means?

I hope you understand WHY that symmetry represents commutativity. Commuted
entries look like a*b and b*a, as marked here, for example:

   *   a   b   c
   a | a |<b>| c |
   b |<b>| c | a |
   c | c | a | b |

These are on opposite sides of the diagonal shown here:

   \   a   b   c
   a | \ |<b>| c |
   b |<b>| \ | a |
   c | c | a | \ |

The symmetry only looks like a perfect geometrical reflection if you make
sure that the cells are square, so that distances are the same
horizontally and vertically:

   *   a   b   c       \   a   b   c
     +---+---+---+       \---+---+---+
   a | a |<b>| c |     a | \ |<b>| c |
     +---+---+---+       +---\---+---+
   b |<b>| c | a |     b |<b>| \ | a |
     +---+---+---+       +---+---\---+
   c | c | a | b |     c | c | a | \ |
     +---+---+---+       +---+---+---\

Does that help?

- Doctor Peterson, The Math Forum 

Date: 06/06/2012 at 10:14:40
From: diegs
Subject: Thank you (Testing for commutativity)

Dr. Peterson,

You are right, sir, I think I misunderstand what symmetry about a line
means. Thank you so much.

Hope someday I could also offer the same help to others as you do.

Associated Topics:
College Modern Algebra

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