We prove that there exists an integer-valued function f on positive integers such that if a finite group G has at most k real-valued irreducible characters, then |G/Sol(G)| is at most f(k), where Sol(G) denotes the largest solvable normal subgroup of G. In the case k = 5, we further classify G/Sol(G). This partly answers a question of Iwasaki [15] on the relationship between the structure of a finite group and its number of real-valued irreducible characters.

中文翻译：

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关于有限群不可约实值字符的个数

我们证明在正整数上存在整数值函数 f 使得如果有限群 G 至多有 k 个实值不可约字符，则 |G/Sol(G)| 至多是 f(k)，其中 Sol(G) 表示 G 的最大可解正态子群。在 k = 5 的情况下，我们进一步分类 G/Sol(G)。这部分回答了 Iwasaki [15] 关于有限群的结构与其实值不可约字符数之间的关系的问题。