On finite groups with an irreducible character large degree

L. S. Kazarin ; S. S. Poiseeva

Modelirovanie i analiz informacionnyh sistem, Tome 22 (2015) no. 4, pp. 483-499.

Résumé

Let $G$ be a finite nontrivial group with an irreducible complex character $\chi$ of degree $d=\chi(1)$. It is known from the orthogonality relation that the sum of the squares of degrees of irreducible characters of $G$ is equal to the order of $G$. N. Snyder proved that if $|G|=d(d+e)$, then the order of $G$ is bounded in terms of $e$, provided $e>1$. Y. Berkovich proved that in the case $e=1$ the group $G$ is Frobenius with the complement of order $d$. We study a finite nontrivial group $G$ with an irreducible complex character $\Theta$ such that $|G|\leq2\Theta(1)^2$ and $\Theta(1)=pq$, where $p$ and $q$ are different primes. In this case we prove that $G$ is solvable groups with abelian normal subgroup $K$ of index $pq$. We use the classification of finite simple groups and prove that the simple nonabelian group whose order is divisible by a prime $p$ and of order less than $2p^4 $ is isomorphic to $L_2(q), L_3(q), U_3(q), Sz(8), A_7, M_{11}$ or $J_1$.

Détail

Keywords: finite group, character of a finite group, irreducible character degree of a finite group.

Bibliographie

[1] Isaacs I. M., Character theory of finite groups, Academic press, New York–San Francisco–London, 1976, 305 pp. | MR | Zbl

[2] Kazarin L. S., Sagirov I. A., “On the degrees of irreducible characters of finite simple groups”, Proc. of the Steklov Inst. Math. Suppl. (Supplementary issues), 2001, suppl. 2, S71–S81 | MR | Zbl

[3] Snyder N., “Groups with a character of large degree”, Proc. Amer. Math. Soc., 136 (2008), 1893–1903 | DOI | MR | Zbl

[4] Berkovich Y., “Groups with few characters of small degree”, J. Math. Israel, 110 (1999), 325–332 | DOI | MR | Zbl

[5] Isaacs I. M., “Bounding the order of a group with a large degree character”, J. Algebra, 348:1 (2011), 264–275 | DOI | MR | Zbl

[6] Durfee C., Jensen S., “A bound on the order of a group having a large character degree”, J. Algebra, 338 (2011), 197–206 | DOI | MR | Zbl

[7] Lewis M. L., “Bounding group order by large character degrees: A question of Snyder”, Journal of Group Theory, 17:6 (2014), 1081–1116 | DOI | MR | Zbl

[8] Feit W., The representation theory of finite groups, North-Holland Publishing Company, Amsterdam–New York–Oxford, 1982, 510 pp. | MR | Zbl

[9] Belonogov V. A., Predstavlenija i haraktery v teorii konechnyh grupp, UrO AN SSSR, Sverdlovsk, 1990 (in Russian) | MR

[10] Bierbrauer J., “The uniformly 3-homogeneus subsets in $PGL(2,q)$”, J. Algebraic Combinatorics, 4 (1995), 99–102 | DOI | MR | Zbl

[11] Walter J., “The characterization of finite groups with abelian Sylow 2-subgroups”, Ann. Math., 8 (1969), 405–514 | DOI | MR

[12] Chabot P., “Groups whose Sylow 2-subgroups have cyclic commutator groups. I; II; III”, J. Algebra, 19 (1971), 21–30 ; 21 (1972), 312–320 ; 29 (1974), 455–458 | DOI | MR | Zbl | DOI | Zbl | DOI | MR | Zbl

[13] Gorenstein D., Finite Simple Groups. An introduction to their classification, Plenum Publishing Corporation, New York, 1982, 333 pp. | MR | Zbl

[14] Conway J. H., Curtis R. T., Norton S. P., Parker R. A., Wilson R. A., Atlas of finite groups: maximal subgroups and ordinary characters for simple groups, Clarendon Press, Oxford, 1985, 255 pp. | MR | Zbl

[15] Kazarin L. S., Poiseeva S. S., “Finite groups with large irreducible character”, Mathematical Notes, 98:2 (2015), 265–272 | DOI | DOI | MR | Zbl

[16] Gorenstein D., Finite Group, Chelsea publishing company, New York, 1968, 522 pp.

[17] Schmidt O. Y., “Groups all whose subgroups are nilpotent”, Mat. Sb., 31 (1924), 366–372 | Zbl

[18] Amberg B., Kazarin L. S., “ABA-groups with cyclic subgroup B”, Trudy instituta matematiki i mekhaniki UrO RAN, 18, no. 3, 2012, 10–22