Intereting Posts

Dual Optimization Problem
finding the number of bit string containing eight 0s and twelve 1s?
Coproducts in $\text{Ab}$
Proving that $S_n$ has order $n!$
Find a general control and then show that this could have been achieved at x2
Subtraction of a negative number
Prove existence of a real root.
$p$-Splittable Integers
Proof of Cauchy-Schwarz inequality – Why select s so that so that $||x-sy||$ would be minimized?
questions about Rudin's summation by parts
Showing that projections $\mathbb{R}^2 \to \mathbb{R}$ are not closed
Use of the Reciprocal Fibonacci constant?
Number of combinations and permutations of letters
De Morgan's law on infinite unions and intersections
A little more on $\sqrt{\cos\bigl(\tfrac{2\pi}7\bigr)}+\sqrt{\cos\bigl(\tfrac{4\pi}7\bigr)}+\sqrt{\cos\bigl(\tfrac{8\pi}7\bigr)}$

Are there any neat examples of non-associative loops such that for each element a in the loop there exists $a^{-1}$ so that $a*a^{-1}=1=a^{-1}*a$. Even cooler would be a commutative loop. Also: are there commutative finite loops?

- Characterization of irreducible elements in integral domains.
- Units in a ring of fractions
- Finite Galois extensions of the form $\frac{\mathbb Z_p}{\langle p(x)\rangle}:\mathbb Z_p$
- Example of a finite non-commutative ring without a unity
- Why don't we have an isomorphism between $R$ and $ R]$?
- About group multiplication table
- How to determine the matrix of adjoint representation of Lie algebra?
- Application of the Sylow Theorems to groups of order $p^2q$
- Endomorphism Rings of finite length Modules are semiprimary
- Appearance of Formal Derivative in Algebra

See also the Parker Loop which is a finite loop of order $2^{13}$ related to the binary Golay code, $M_{24}$ (largest sporadic Mathieu group), Conway’s construction of the Monster group, etc.

I think most of your questions are answered by looking at Moufang loops.

A loop in which the left and right inverse agree (a loop with two-sided inverses) is called an IP-loop. Sometimes people replace a loop by an isotope, which basically scrambles and relabels the multiplication table (apply a row and column permutation, and a permutation of the underlying set). For groups, that would basically be crazy, but loops are not terribly messed up by such an operation.

A loop is a Moufang loop iff every isotope has two-sided inverses.

Non-associative, commutative, Moufang loops have order a multiple of 81, and there are two non-isomorphic such loops. They were constructed by M. Hall Jr.

Zassenhaus’s Commutative Moufang Loop is an example of commutative loop of order $81$ which is not a group.

- Is every monotone map the gradient of a convex function?
- Why is the ring of matrices over a field simple?
- Help find hard integrals that evaluate to $59$?
- half-derivative of $x^2$
- Do De Morgan's laws hold in propositional intuitionistic logic?
- Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$
- Understanding regular matrices
- Homotopy of singular $n$-simplices
- Can the semidirect product of two groups be abelian group?
- How can I prove that the binomial coefficient ${n \choose k}$ is monotonically nondecreasing for $n \ge k$?
- How to prove $\cos\left(\pi\over7\right)-\cos\left({2\pi}\over7\right)+\cos\left({3\pi}\over7\right)=\cos\left({\pi}\over3 \right)$
- Decomposition of semimartingales
- Homomorphisms from $S_4$ to $\mathbb Z_2$
- Is there a hyperbolic geometry equivalent to Möbius transformations in spherical geometry?
- What is the difference between Completeness and Soundness in first order logic?