On the partitions of finite \(p\)-groups. II (Q1192778)

[For part I cf. ibid. VIII. Ser. 82, 1-5 (1988; Zbl 0677.20016).] Given a finite group \(G\), a partition of \(G\) is a set \(\pi = \{H_ 1, \dots, H_ t\}\) of subgroups of \(G\) such that \(G-\{1\}\) is the disjoint union of the sets \(H_ i - \{1\}\). If \(\pi = \{G\}\) the partition is trivial and it is abelian if each \(H_ i\) is abelian. If \(| G| = p^ n\), a partition of \(G\) is of typical order \(p^ k\) if there exists an integer \(k\) such that at least \(p^{n - k}\) elements of the partition are of order \(p^ k\). In this paper it is proved that if \(G\) is a group of order \(p^ n\) and exponent \(p\), having a nontrivial partition of typical order greater or equal to \(p^ k\), \(k \geq 2\), then the minimum of the set \(\{\chi(1) \mid \chi \in \text{Irr} (G)\), \(\chi(1) \neq 1\}\) is lower bounded by \(p^ 2\).