[Maths-Education] RE: Group Theory and Soduko

[Ernest, Paul]

Dear Colleagues

My Group theory is a bit rusty, but I was looking at a completed Soduko  which is a 9X9 grid and it is obviously a permutation - each line is a rearrangemnt of previous with no elements staying in same place - so my thought is - it should (or could?) be a permutation group of order 9. Now I only recall 2 finite groups of order 9 - C9 (cyclic group of order nine) and C3XC3 (cartesian product of 2 cyclic groups of order 3)

But playing with transformations/rearrangements of the Soduko square did not get me anywhere.

The number of nine element permutations is 9! so the 9 shown in an S grid is a tiny part of all possible perms - do these form a group? If so, what operation transforms them into each other?

The answer may differ for different completed Sodukos as they constitute different selections of 9 permutations

To be a group, one of them has to be a unit transformation, and each needs an inverse

Actually I'm starting to have doubts as to where an aribtrary Soduko grid does form a group

I'm sure someone else has thought about this and can answer it better than me!

More generally, how do you determine a group from a finite group table (without the labels)

Any interest or answers?

Best wishes

Paul
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Paul Ernest
Emeritus Professor
University of Exeter
SELL, St Lukes, Heavitree Road
Exeter  EX1 2LU, UK

Visiting Professor, HiST-ALT, Norway
Visiting Professor, UiO, Norway

http://www.people.ex.ac.uk/PErnest/
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