On Finite Groups with an Irreducible Character Large Degree
https://doi.org/10.18255/1818-1015-2015-4-483-499
Abstract
Let G be a finite nontrivial group with an irreducible complex character χ of degree d = χ(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 Θ such that |G| ≤ 2Θ(1)2 and Θ(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 2p4 is isomorphic to L2(q), L3(q), U3(q), Sz(8), A7, M11 or J1.
About the Authors
L. S. Kazarin
P.G. Demidov Yaroslavl State University
Russian Federation
Russian Federation
doctor of science, professor
Sovetskaya str., 14, Yaroslavl, 150000, Russia
S. S. Poiseeva
P.G. Demidov Yaroslavl State University
Russian Federation
Russian Federation
graduate student
Sovetskaya str., 14, Yaroslavl, 150000, Russia
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Review
For citations:
Kazarin L.S., Poiseeva S.S. On Finite Groups with an Irreducible Character Large Degree. Modeling and Analysis of Information Systems. 2015;22(4):483-499. (In Russ.) https://doi.org/10.18255/1818-1015-2015-4-483-499
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