Wreath products and representations of degree one or two

Authors:
J. M. Bateman, Richard E. Phillips and L. M. Sonneborn

Journal:
Trans. Amer. Math. Soc. **181** (1973), 143-153

MSC:
Primary 20F25

MathSciNet review:
0320158

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Abstract | References | Similar Articles | Additional Information

Abstract: denotes all groups 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 and is wr in (1) has been completely solved by D. Parker in the case where and 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 is a finite -group: If is a finite -group, wr is in iff is finite and is isomorphic to a subgroup of a dihedral group of an elementary -group. If is not a -group, we offer only necessary conditions on . Problem (1) is closely related to Problem (2): If is a prime field or the integers, which finite groups have all their irreducible representations over of degrees one or two? It is shown that all finite which satisfy (2) are groups; in particular all such are solvable.

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DOI:
https://doi.org/10.1090/S0002-9947-1973-0320158-0

Keywords:
Supersolvable,
solvable wreath product,
group ring,
modular representation,
integral representation

Article copyright:
© Copyright 1973
American Mathematical Society