# How many? -question

by Zaare
Tags: how manyquestion
 P: 54 The problem: Let $$G$$ be a set with an associative binary operation and $$e \in G$$ an element satisfying the following conditions: 1) $$eg=g$$ for any $$g \in G$$. 2) For any $$g$$ there is $$h$$ such that $$gh=e$$. Assume that $$p$$ is a prime number and $$G$$ has $$p$$-elements. How many non isomorphic such binary operations are on $$G$$ which are not groups? I know that there are 2 such operations if $$\mid G \mid =p^2$$, and 3 such operations if $$\mid G \mid =p*q$$, if $$p$$ and $$q$$ 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.