(adsbygoogle = window.adsbygoogle || []).push({}); "How many?"-question

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# How many? -question

**Physics Forums | Science Articles, Homework Help, Discussion**