Some behaviors of FSZ groups under central products, central quotients, and regular wreath products

Abstract

We show that any group G with a non-F SZm quotient by a central cyclic subgroup also provides a non-F SZm group of order m|G| obtained as a central product of G with a cyclic group. We then construct, for every prime p > 3 and j ∈ N, an F SZ p j group F such that there is a central cyclic subgroup A with F/A not F SZ p j . We apply these results to regular wreath products to construct an F SZ p-group which is not F SZ + for any prime p > 3. These give the first known examples of F SZ groups that are not F SZ + . We are also able to prove a few partial results concerning the F SZ properties for the Sylow subgroups of symmetric groups. In the appendix we enumerate all non-F SZ groups of order 5 7 .

Automorphisms of Kronrod–Reeb graphs of Morse functions on compact surfaces

“…Remark 3.3. Wreath products D Z p , where p is a simple, and D is a p-group, for example, the groups Z p Z p , (Z p Z p ) Z p , were studied in [10], [12,Theorems 4.7,5.5] in connection with the so-called FSZ-property. Proof.…”

A family of non-FSZ finite symplectic groups

“…The author has constructed several other families of F SZ and non-F SZ p-groups for p > 3 in[12,13]. Whether or not non-F SZ 2-groups or 3-groups exist remains an open question.…”