Wreath products and representations of degree one or two

Abstract:

${\mathcal {S}_2}$ denotes all groups $G$ that possess an ascending invariant series whose factors are one- or two-generated Abelian groups. We are interested in the ptoblem (1): For which nontrivial groups $A$ and $B$ is $A$ wr $B$ in ${\mathcal {S}_2}?$ (1) has been completely solved by D. Parker in the case where $A$ and $B$ are finite of odd order. Parker’s results are partially extended here to cover groups of even order. Our answer to (1) is complete in the case where $A$ is a finite $2$-group: If $A$ is a finite $2$-group, $A$ wr $B$ is in ${\mathcal {S}_2}$ iff $B$ is finite and $B/{O_2}(B)$ is isomorphic to a subgroup of a dihedral group of an elementary $3$-group. If $A$ is not a $2$-group, we offer only necessary conditions on $B$. Problem (1) is closely related to Problem (2): If $F$ is a prime field or the integers, which finite groups $B$ have all their irreducible representations over $F$ of degrees one or two? It is shown that all finite $B$ which satisfy (2) are ${\mathcal {S}_2}$ groups; in particular all such $B$ are solvable.