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Groups with Representations of Bounded Degree

  • I. M. Isaacs (a1) and D. S. Passman

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Let G be a discrete group with group algebra C[G] over the complex numbers C. In (5) Kaplansky essentially proves that if G has a normal abelian subgroup of finite index n, then all irreducible representations of C[G] have degree ≤n. Our main theorem is a converse of Kaplansky's result. In fact we show that if all irreducible representations of C[G] have degree ≤n, then G has an abelian subgroup of index not greater than some function of n. (The degree of a representation of C[G] for arbitrary G is defined precisely in § 3.)

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References

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1. Amitsur, S. A., Groups with representations of bounded degree II, Illinois J. Math., 5 (1961), 198205.
2. Curtis, C. W. and Reiner, I., Representation theory of finite groups (New York, 1962).
3. Feit, W. and Thompson, J. G., Groups which have a faithful representation of degree less than (p - l)/2, Pac. J. Math., 11 (1961), 12571262.
4. Jacobson, N., Structure of rings, Amer. Math. Soc. Coll. Pub. vol. 37.
5. Kaplansky, I.. Groups with representations of bounded degree, Can. J. Math., 1 (1949), 105 112.
6. Kurosh, A. G., The theory of groups, Vol. II (New York, 1960).
7. Passman, D. S., On groups with enough finite representations, Proc. Amer. Math. Soc. 14 (1963), 782787.
8. Rickart, C., Uniqueness of norm in Banach algebras, Ann. of Math., 51 (1950), 615628.
9. Speiser, A., Die Théorie der Gruppen von endlicher Ordnung (Basel, 1956).
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Groups with Representations of Bounded Degree

  • I. M. Isaacs (a1) and D. S. Passman

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