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Suppose $S$ is a set of groups of order $n$, is there a binary operation $*:S\times S \to S$ that is definable on $S$?

The obvious operations I started with were

  • Cartesian product, but that produces groups of order $n^2$.
  • Intersection, but that produces groups of order $\leq n$.
  • Matrix multiplication of the Cayley table representations, but that doesn't produce another $n\times n$ matrix over the integers $\{0,1,...\,n-1\}$

I'm curious, are there are any known operations one can define that takes pairs of groups and produces one of the same order?

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    But what does that mean? What's the intersection of, say, $S_3$ and $\mathbb{Z} / 6 \mathbb{Z}$?2011-10-26

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Assuming isomorphism classes of groups is intended, the answer to the question is yes, because we can just make the set of $|S|$ groups of order $n$ into a cyclic group of order $|S|$ in some arbitrary fashion. But if you mean is there some "natural" way of doing it that involves the group structure of the groups in $S$, then the answer is almost certainly no.

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    I will look for some structural feature that gives a unique integer between 0 and n-1, this sounds like a straightforward a$p$$p$roach.2011-10-27