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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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: $ {\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.


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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