Thread: Order of Multiplication for Symmetry Tables describing Geometric Shapes

I am construction multiplication tables for the symmetries of various geometric figures and I find myself wondering if there is a standardised order in which the tables are meant to be read?

__ | D |_ F^2_ |

r __|__ |_ #_ |

r^2|___ |___ |

(The values D, F^2,... are meaningless so ignore them.)

My question is which goes first, the row or the column operation? Is the position denoted by # in the table above r*F^2 = # or F^2*r = #. Or is the order table specific? (This seems unlikely.)

The PDF I'm using implies (Row*Column; or a left to right bias as when reading english) but doesn't explicitly say that. I am just looking for confirmation of that though.

Sorry for the super simple question. I did a search and it only returned things that looked unrelated.

Hello.

You are correct: The order of factors is always row*column, so that # = r*F^2.

it is a convention, which largely hinges on how you interpret ab, when a and b are functions.

some authors write mappings on the left (for example, herstein), in which case ab means "first do a, then do b".

others (for example, fraleigh, i believe) write ab to mean "first do b, then do a".

the difference is purely one of orientation, just like the convention in English is to read left-to-right. other languages are different.

if you think of the table as being a matrix, it is the difference between a matrix and its transpose.

the "usual" convention, according to "most" authors, is as you surmise, and HappyJoe confirms. just be aware,

the some people do the opposite, which can lead to some confusion.