Commutative automorphic loops of order p^3 [article]

Dylene Agda Souza de Barros, Alexander Grishkov, Petr Vojtěchovský
2015 arXiv   pre-print
A loop is said to be automorphic if its inner mappings are automorphisms. For a prime p, denote by A_p the class of all 2-generated commutative automorphic loops Q possessing a central subloop Z Z_p such that Q/Z Z_p× Z_p. Upon describing the free 2-generated nilpotent class two commutative automorphic loop and the free 2-generated nilpotent class two commutative automorphic p-loop F_p in the variety of loops whose elements have order dividing p^2 and whose associators have order dividing p, we
more » ... show that every loop of A_p is a quotient of F_p by a central subloop of order p^3. The automorphism group of F_p induces an action of GL_2(p) on the three-dimensional subspaces of Z(F_p) ( Z_p)^4. The orbits of this action are in one-to-one correspondence with the isomorphism classes of loops from A_p. We describe the orbits, and hence we classify the loops of A_p up to isomorphism. It is known that every commutative automorphic p-loop is nilpotent when p is odd, and that there is a unique commutative automorphic loop of order 8 with trivial center. Knowing A_p up to isomorphism, we easily obtain a classification of commutative automorphic loops of order p^3. There are precisely 7 commutative automorphic loops of order p^3 for every prime p, including the 3 abelian groups of order p^3.
arXiv:1509.05727v1 fatcat:2zm7veljnrahxaw6z5ar7p7sme