Aleš Drápal

Left conjugacy closed loops of nilpotency class two

Every LCC loop Q with Inn Q abelian is nilpotent class two. A loop Q of nilpotency class two is LCC ⇔ L(x, y) = L(y, x) for all x, y ∈ Q ⇔ ℑ/Z(Mlt Q) is abelian ⇔ [x, y, z] = [x,z,y] for all x, y, z ∈ Q ⇔ [x, y, z] = [xy, z][x, z]−1 for all x, y, z ∈ Q. All nilpotent LCC loops of order p2 are described, and some of their multiplication groups are computed.

Homomorphisms of primitive left distributive groupoids

Let Ak(*) denote the left distributive groupoid defined on such that mod2k for every a ∈ Ak For 1≥k<n define so that For every a ∈ Ak Let r be the greatest integer such that 2r divides n – k. Then σk,n is a groupoid homomorphism iff k ≥ 2r+1.

On left conjugacy closed loops with a nucleus of index two

A loop Q is said to be left conjugacy closed (LCC) if the left translations form a set of permutations that is closed under conjugation. Loops in which the left and middle nuclei coincide and are of index 2 are necesarilly LCC, and they are constructed in the paper explicitly. LCC loops Q with the right nucleus G of index 2 offer a larger diversity, but that is associated with the level of commutativity of G (amongst others, the centre of G has to be nontrivial). On the other hand, for each m ≥ 2 one can construct an LCC loop Q of order 2m in such a way that its left nucleus is trivial, and the right nucleus if of order m. If Q is involutorial, then it is a Bol loop.

Non-isomorphic 2-groups Coincide at Most in Three Quarters of their Multiplication Tables

Let G(?) and G(*) be two groups of the finite order n, and let d be the size of the set {(a, b) ?G×G;a?b?=a*b }. Let P and Q be Sylow 2-subgroups of G(?) and G(*), respectively. If d is less than n2/4, then there exists an isomorphism ?: P?Q, and the normalizers of P and Q have the same order.

Finite left distributive algebras with one generator

Abstract Finite monogenerated groupoids G satisfying the left distributive law x · ( y · z ) = ( x · y ) · ( x · z ) are studied. They are shown to reduce over q G = {(a, b) ϵ G 2 ; ca = cb for all c ϵ G} and p G = {(a, b) ϵ G 2 ; ac = bc for all c ϵ G} to a groupoid isomorphic to A k = A k (∗), k ≥ 0 . ( A k is the unique left distributive groupoid on {1, …, 2 k } with a ∗ 1  a + 1 mod 2 k for every 1 ≤ a ≤ 2 k .) G ≅ A k is proved to hold whenever b → a · b equals id G for some a ϵ G . We describe all cases when G = Ga ∪ { b } for some a , b ϵ G , and all cases when there exists a binary operation o on G such that G (·, o ) satisfies the axioms of left distributive algebras.

How far apart can the group multiplication tables be

Put dist(G(·), G(*)) = card{(a, b) eG2; a · b ≠ a * b} for any two groups G(·), G(*) with the same underlying set and δ(G(·)) = min dist(G(·), G(*)), where G(*) runs through all groups with dist(G(·), G(*)) ≠ 0. It holds that δ(G(·)) e {6n − 24, 6n − 20, 6n − 18} for any n ⩾ 51, n being the order of G. Moreover, groups G(·) and G(*) are isomorphic whenever dist(G(·), G(*)) ⩽ n2/9.

Structural interactions of conjugacy closed loops

We study conjugacy closed loops by means of their multiplication groups. Let Q be a conjugacy closed loop, N its nucleus, A the associator subloop, and L and R the left and right multiplication groups, respectively. Put M = {a ∈ Q; L a ∈R}. We prove that the cosets of A agree with orbits of [L,R], that Q/M ≅ (Inn Q) L 1 and that one can define an abelian group on Q/N x Mlt 1 . We also explain why the study of finite conjugacy closed loops can be restricted to the case of N/A nilpotent. Group [L,R] is shown to be a subgroup of a power of A (which is abelian), and we prove that Q/N can be embedded into Aut([L, R]). Finally, we describe all conjugacy closed loops of order pq.

Alternating groups and Latin squares

For every n ⩾ 1, n ≠ 2, 4, 5, the alternating group A ( n ) on n symbols is equal to the multiplication group of a loop.

On Left Conjugacy Closed Loops in which the Left Multiplication Group is Normal

A loopQ is said to be left conjugacy closed (LCC) if the left translations form a set of permutations that is closed under conjugation. This papers investigates those LCC loops where the group generated by left translations is normal in the group generated by both left and right translations.

Persistence of cyclic left distributive algebras

Abstract Let A k = A k (∗) denote the left distributive groupoid on {0, 1, …, 2k − 1} such that a ∗ 1  a + 1 mod 2 k for every a ϵ Ak. Let d ≥ 0 and put r = max {i; 2i divides d}. For a = ∑ ai2i ϵ Ak, ai ϵ {0, 1}, put νd(a) = ∑ aiνd(2i) and vd(2i) = 2(i + 1) 2d − 2i2d. Then vd: Ak → Ak2d is a groupoid homomorphism iff k ≤ 22r + 1.