On Finite Groups with an Irreducible Character Large Degree
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...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2015-08-01
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| Series: | Моделирование и анализ информационных систем |
| Subjects: | |
| Online Access: | https://www.mais-journal.ru/jour/article/view/267 |
| Summary: | 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. |
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| ISSN: | 1818-1015 2313-5417 |