Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

On the solvability of unit groups of group algebras


Author: J. M. Bateman
Journal: Trans. Amer. Math. Soc. 157 (1971), 73-86
MSC: Primary 20.80
DOI: https://doi.org/10.1090/S0002-9947-1971-0276371-2
MathSciNet review: 0276371
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let FG be the group algebra of a finite group $ G$ over a field $ F$ of characteristic $ p \geqq 0$; and let $ U$ be the group of units of FG. We prove that $ U$ is solvable if and only if (i) every absolutely irreducible representation of $ G$ at characteristic $ p$ is of degree one or two and (ii) if any such representation is of degree two, then it is definable in $ F$ and $ F = GF(2)$ or $ GF(3)$. This result is translated into intrinsic group-theoretic and field-theoretic conditions on $ G$ and $ F$, respectively. Namely, if $ {O_p}(G)$ is the maximum normal $ p$-subgroup of $ G$ and $ L = G/{O_p}(G)$, then (i) $ L$ is abelian, or (ii) $ F = GF(3)$ and $ L$ is a $ 2$-group with exactly $ (\vert L\vert - [L:L'])/4$ normal subgroups of index 8 that do not contain $ L'$, or (iii) $ F = GF(2)$ and $ L$ is the extension of an elementary abelian $ 3$-group by an automorphism which inverts every element.

Conditions are found for the nilpotency, supersolvability, and $ p$-solvability of $ U$.


References [Enhancements On Off] (What's this?)

  • [1] E. Artin, Geometric algebra, Interscience, New York, 1957. MR 18, 553. MR 0082463 (18:553e)
  • [2] J. M. Bateman and D. B. Coleman, Group algebras with nilpotent unit groups, Proc. Amer. Math. Soc. 19 (1968), 448-449. MR 36 #5238. MR 0222186 (36:5238)
  • [3] M. Burrow, Representation theory of finite groups, Academic Press, New York, 1965. MR 38 #250. MR 0231924 (38:250)
  • [4] D. B. Coleman, On the modular group ring of a $ p$-group, Proc. Amer. Math. Soc. 15 (1964), 511-514. MR 29 #2306. MR 0165015 (29:2306)
  • [5] C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure and Appl. Math., vol. 11, Wiley, New York, 1962. MR 26 #2519. MR 0144979 (26:2519)
  • [6] W. E. Deskins, Finite abelian groups with isomorphic group algebras, Duke Math. J. 23 (1956), 35-40. MR 17, 1052. MR 0077535 (17:1052c)
  • [7] M. Hall, Jr., The theory of groups, Macmillan, New York, 1959. MR 21 #1996. MR 0103215 (21:1996)
  • [8] L. K. Hua, On the multiplicative groups of a field, Acad. Sinica Sci. Record 3 (1950), 1-6. MR 12, 584. MR 0039707 (12:584e)
  • [9] J. Rotman, The theory of groups. An introduction, Allyn and Bacon, Boston, Mass., 1965. MR 34 #4338. MR 0204499 (34:4338)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20.80

Retrieve articles in all journals with MSC: 20.80


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0276371-2
Keywords: Group algebra, unit group, radical, separable algebra, solvability, irreducible representations
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society