Left conjugacy closed loops of nilpotency class two

Code loops in both parities

“…(This also means that the unpleasant sign change in C(u, v) = Power-associativity of odd code loops for p = 3 is an artifact of combinatorial polarization, and it has to be explicitly enforced in Definitions 3.2-3.4 (by the assumptions u i = 0, [x, x, x] = 1, and f is characteristic, respectively). It is perhaps not obvious that the condition [x, x, x] = 1 is independent of the remaining assumptions in Definition 3.3, but the following example shows that it is: The following example shows that (10.3) is independent of the remaining conditions in Definition 3.1: Example 10.3 As in [7], let (Q, * ) be defined on Z 25 by x * y = x +y +5x 2 y. Then Q is a symplectic conjugacy closed loop in which x (5) = 1 does not hold for all x ∈ Q.…”

Schreier loops

“…These loops are centrally nilpotent of nilpotency class two, which means that the factor loop L/N is abelian. Some examples of such loops are given in [5] p. 36, [18], Remark 17.6, [7], Section 5, [14], Lemma Let N and K be abelian groups and let T (σ) = Id for all σ ∈ K. Then the multiplication in the loop L(Id, f) is given by…”

“…[20], [21], [16], Chapter XII, §48-49, pp. 121-131, [24], Chapter 2,§7,. Any group which is an extension G of a normal subgroup N by a group K is determined by two identities describing the action of K on N and ensuring the associativity of the extension G by the choice of a system of representatives in G for the factor group G/N isomorphic to K.…”

Bruck Loops with Abelian Inner Mapping Groups

“…automorphic loops [20], left conjugacy closed loops [8], and 2-divisible Moufang loops [19]. It is the purpose of this paper to lengthen this list.…”