Is satisfying Lagrange's theorem sufficient to show a loop is a group? Consider a finite loop $L$, that is a set with an operation that has closure, identity, and invertibility (but not necessarily associativity). Consider a subloop of $L$, that is a set $H$ contained in $L$ that is itself a loop (with the same identity element). If the order of all subloops of $L$ divide the order of $L$ ("satisfies Lagrange's theorem"), is the loop $L$ necessarily a group?
Can this be proven or is there an easy counterexample of a finite loop satisfying Lagrange's theorem that is not a group?
 A: No.
Consider the loop $Q^*$ obtained from the hyperbolic quaternions: The set $\{ \pm1, \pm i, \pm j, \pm k \}$ with the loop operation being multiplication defined by $i^2=j^2=k^2= +1$ and the other products just as in the quaternions. You can verify that this is clearly a loop but not a group. Associativity is not satisfied: $ i(jj)=i \not = -i = (ij)j$.
The subloops of $Q^*$ are $\{1\}, \{1,-1\} \{1,g\} \{1,g,-g,-1\}$ where $g \not = \pm 1$ and $Q^*$. Clearly, the orders of all the subloops divide $8$.
A: The smallest example is one of the non-associative loops of order 5. Its Cayley table can be displayed as:
01234
12043
23401
34120
40312

Since each row and column contains each number exactly once and since the first row and first column are the identity, this is a loop. Since $1 \cdot (1 \cdot 1) = 1 \cdot 2 = 0 \neq (1 \cdot 1) \cdot 1 = 2 \cdot 1 = 3$ this loop is not even power-associative (it is the unique loop of order 5 that is not power associative).
The subloop generated by each nonzero element is the entire loop, so the only subloops are order 1 (just the zero element) and order 5 (the whole loop).
Since every loop of order 4 or smaller is associative (a group), this is an example of minimal size. I just checked the other loops of order 5 and they have subloops of size 2, so this is the unique up to isomorphism non-associative loop of minimal size satisfying Lagrange's theorem.
I learned everything I know about loops from Stephen Gagola III. I recommend his articles, especially on Moufang loops (all of which satisfy Lagrange's theorem). The proof of that result mirrors a lot of finite group theory, including proceeding through odd order, solvable, and a classification of finite simple moufang loops as being basically PSL groups.
