Small Loops of Nilpotency Class Three with Commutative Inner Mapping Groups

Groups with commuting inner mappings are of nilpotency class at most two, but there exist loops with commuting inner mappings and of nilpotency class higher than two, called loops of Csörgő type. In order to obtain small loops of Csörgő type, we expand our programme from Explicit constructions of loops with commuting inner mappings, European J. Combin. 29 (2008), 1662–1681, and analyze the following setup in groups: Let G be a group, Z ≤ Z(G), and suppose that δ : G/Z × G/Z → Z satisfies δ(x, x) = 1, δ(x, y) = δ(y, x)−1, zδ([z, y], x) = zδ([z, x], y) for every x, y, z ∈ G, and δ(xy, z) = δ(x, z)δ(y, z) whenever {x, y, z} ∩G′ is not empty. Then there is μ : G/Z × G/Z → Z with δ(x, y) = μ(x, y)μ(y, x)−1 such that the multiplication x ∗ y = xyμ(x, y) defines a loop with commuting inner mappings, and this loop is of Csörgő type (of nilpotency class three) if and only if g(x, y, z) = δ([x, y], z)δ([y, z], x)δ([z, x], y) is nontrivial. Moreover, G has nilpotency class at most three, and if g is nontrivial then |G| ≥ 128, |G| is even, and g induces a trilinear alternating form. We describe all nontrivial setups (G,Z, δ) with |G| = 128. This allows us to construct for the first time a loop of Csörgő type with an inner mapping group that is not elementary abelian.