Wreath products and representations of degree one or two

Bateman, J. M.; Phillips, Richard E.; Sonneborn, L. M.

doi:10.1090/S0002-9947-1973-0320158-0

Wreath products and representations of degree one or two
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by J. M. Bateman, Richard E. Phillips and L. M. Sonneborn PDF

Trans. Amer. Math. Soc. 181 (1973), 143-153 Request permission

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.

References

J. M. Bateman, On the solvability of unit groups of group algebras, Trans. Amer. Math. Soc. 157 (1971), 73–86. MR 276371, DOI 10.1090/S0002-9947-1971-0276371-2
Roger Carter and Paul Fong, The Sylow $2$-subgroups of the finite classical groups, J. Algebra 1 (1964), 139–151. MR 166271, DOI 10.1016/0021-8693(64)90030-4
Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
John R. Durbin, Finite supersolvable wreath products, Proc. Amer. Math. Soc. 17 (1966), 215–218. MR 186734, DOI 10.1090/S0002-9939-1966-0186734-5
K. W. Gruenberg, Residual properties of infinite soluble groups, Proc. London Math. Soc. (3) 7 (1957), 29–62. MR 87652, DOI 10.1112/plms/s3-7.1.29
B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703, DOI 10.1007/978-3-642-64981-3
Hanna Neumann, Varieties of groups, Springer-Verlag New York, Inc., New York, 1967. MR 0215899, DOI 10.1007/978-3-642-88599-0
Donald B. Parker, Wreath products and formations of groups, Proc. Amer. Math. Soc. 24 (1970), 404–408. MR 252519, DOI 10.1090/S0002-9939-1970-0252519-5
W. R. Scott and L. M. Sonneborn, Supersolvable wreath products, Math. Z. 92 (1966), 154–163. MR 191951, DOI 10.1007/BF01312434
D. A. Suprunenko, Razreshimye i nil′potentnye lineĭ nye gruppy, Izdat. Belgosuniv. im. V. I. Lenina, Minsk, 1958 (Russian). Ministerstvo Vysšego Obrazovanija SSSR. MR 0154917