Posted by: Dave Richeson | February 11, 2009

Is this the Cayley table of a group? (Part 1)

A math major in our department business major asked an interesting question the other day. The student’s abstract algebra class was working with Cayley tables. The class was given a table for a binary operation {}* and was asked to check if it was a group.

Given a table it is easy to check that the operation is closed, that there is a two-sided identity, and that every element has a two-sided inverse. Checking for associativity can be a hassle. For example, in a 5×5 table there are 125 equations of the form a*(b*c)=(a*b)*c to check, thus requiring 500 products to look up. (Of course if the proposed group is a subset of a known group, then checking associativity is not necessary.)

In this case the table happened to be a Latin square—that is, each symbol occurred exactly once in each row and each column. The student wanted to know if that was sufficient to guarantee associativity.

That is, suppose you are given a table for a binary operation such that

  1. there is a two-sided identity,
  2. every element has a two-sided inverse, and
  3. the table is a Latin square.

Is it the Cayley table of a group? If so, prove it.  If not, find a minimal counter-example. (OK, those last two questions are mine.)

I’ll post the solution in a follow-up post.

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  1. [...] for an exam I’m giving next week in my Modern Algebra class and stumbled on an interesting post regarding Cayley tables.  I find Cayley tables very interesting since they’re related to things like “magic [...]



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