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I'm asking here because no textbook or website that I know of gives a definition of the above mentioned term. Since there's no obvious way (that I can think of) to define a normal subloop, I don't see how the definition of a simple group can be modified for the case of loops.

So - what is the definition of a simple loop? (And what is the motivation behind the definition?)

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Where did you encounter the term, first of all? – J. M. Nov 23 '10 at 15:35
I assume it's a loop with no nontrivial quotient loops. (That is, any surjective loop morphism out of the loop must either be an isomorphism or map to the trivial loop.) But who knows? – Qiaochu Yuan Nov 23 '10 at 15:36
@J.M. - In the book "An introduction to quasigroups and their representations". Specifically, they mention that the Moufang loops M(q) are simple. – Pandora Nov 23 '10 at 15:57
up vote 7 down vote accepted

A subloop $H$ of a loop $L$ is a normal subloop if $H$ is the kernel of a homomorphism from $L$ onto some loop $K$, and $L$ is a simple loop if it has no nontrivial normal subloops; that is, the only homomorphic images of $L$ have order $1$ or are isomorphic to $L$. (This is Qiaochu's answer above).

I took the definition out of the review (MR0977475 (90b:20057)) of Bannai, Eiichi; Song, Sung-Yell, The character tables of Paige's simple Moufang loops and their relationship to the character tables of ${\rm PSL}(2,q)$. Proc. London Math. Soc. (3) 58 (1989), no. 2, 209–236,

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Without referencing homomorphisms, a normal subloop K is one such that as sets:

$$\begin{align} xK &= Kx \\ (xy)K &= x(yK) \\ K(xy) &= (Kx) y \end{align}$$

Each of these can also be read as $\forall x, y, k \, \exists k'(x, y, k)$ such that $xk = k'x$, etc. The first equation is just the normality condition for groups. The second two assert that "next to" elements of the normal subgroup, associativity holds up to choosing new elements from the normal subloop. This is enough to get Lagrange's theorem to hold, and for right and left cosets to be equal.

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