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The problem:

Let [tex]G[/tex] be a set with an associative binary operation and [tex]e \in G[/tex] an element satisfying the following conditions:

1) [tex]eg=g[/tex] for any [tex]g \in G[/tex].

2) For any [tex]g[/tex] there is [tex]h[/tex] such that [tex]gh=e[/tex].

Assume that [tex]p[/tex] is a prime number and [tex]G[/tex] has [tex]p[/tex]-elements. How many non isomorphic such binary operations are on [tex]G[/tex] which are not groups?

I know that there are 2 such operations if [tex]\mid G \mid =p^2[/tex], and 3 such operations if [tex]\mid G \mid =p*q[/tex], if [tex]p[/tex] and [tex]q[/tex] are not equal.

So I'm guessing there answer to the problem is 1. What I've been trying to do for the past week has been to show that 1 such operations exists and to find a contradiction by assuming that a second operation also exists.

But I haven't even been able to prove the existence.

Any help would be greatly appreciated.

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# How many? -question

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