Smallest Degrees of Representations of Exceptional Groups of Lie Type

Lübeck, Frank

doi:10.1081/agb-100002175

Cited by 45 publications

(46 citation statements)

References 14 publications

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“…Thus, if r denotes the rank of G, we clearly have |T | ≤ (q + 1) r and hence |L| ≤ (q + 1) r |W | . Comparing this bound with the lower bound for minimal non-trivial character degrees in [10] and using Lemma 2.3, we immediately find that the assertion holds if q ≥ 4. Thus, there remain only finitely many cases to check.…”

Section: Groups Of Lie Type In Non-defining Characteristicmentioning

confidence: 85%

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Finite groups with minimal 1-PIM

Malle

Weigel

2008

manuscripta math.

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Abstract. Let F be a field of characteristic > 0 and let G be a finite group. It is well-known that the dimension of the minimal projective cover Φ G 1 (the so-called 1-PIM) of the trivial left F[G]-module is a multiple of the -part |G| of the order of G. In this note we study finite groups G satisfying dim F (Φ G 1 ) = |G| . In particular, we classify the non-abelian finite simple groups G and primes satisfying this identity (Thm. A). As a consequence we show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers (Cor.B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number ∈ {2, 3, 5} implies the existence of an -Hall subgroup for G (Thm. C). An important tool in our proofs is the super-multiplicativity of the dimension of the 1-PIM over short exact sequences (Prop. 2.2).

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Section: Groups Of Lie Type In Non-defining Characteristicmentioning

confidence: 85%

“…We start with exceptional groups of Lie type. Using Lusztig's classification of complex irreducible characters, Lübeck [10] has determined the smallest nontrivial character degrees of the simple exceptional groups of Lie type. With this result we obtain: Theorem 4.1.…”

Section: Groups Of Lie Type In Non-defining Characteristicmentioning

confidence: 99%

Finite groups with minimal 1-PIM

Malle

Weigel

2008

manuscripta math.

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“…The results obtained have proved very useful in various applications. The irreducible complex characters of finite Lie-type groups of small degrees were studied in [12,16] and more recently in [15]. In these papers, the irreducible complex characters of degrees up to a certain bound are classified and it is proved that there is a large gap between the degrees of these characters and the next degree.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Conjugacy classes of small sizes in the linear and unitary groups

Burkett

Nguyen

2013

Journal of Group Theory

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Abstract. Using the classical results of G. E. Wall on the parametrization and sizes of (conjugacy) classes of finite classical groups, we present some gap results for the class sizes of the general linear groups and general unitary groups as well as their variations. In particular, we identify the classes in GL n .q/ of size up to q 4n 8 and classes in GU n .q/ of size up to q 4n 9 . We then apply these gap results to obtain some bounds and limits concerning the zeta-type function encoding the conjugacy class sizes of these groups.

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“…Diesen Fall gibt es aber nicht (vql. [20] und [13]). Bleibt also noch die Möglichkeit, dass der Charakter den Grad 2p a hat.…”

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