Bounding the number of p'-degree characters from below

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Abstract Let G be a finite group of order divisible by a prime p and let P ∈ Syl p ⁡ ( G ) {P\in{\operatorname{Syl}}_{p}(G)} . We prove a recent conjecture by Hung stating that | Irr p ′ ⁡ ( G ) | ≥ exp ⁡ ( P / P ′ ) - 1 p - 1 + 2 ⁢ p - 1 - 1 {|{\operatorname{Irr}}_{p^{\prime}}(G)|\geq\frac{\exp(P/P^{\prime})-1}{p-1}+2% \sqrt{p-1}-1} . Let a ≥ 2 {a\geq 2} be an integer and suppose that p a {p^{a}} does not exceed the exponent of the center of P. We also show that the number of conjugacy classes of elements of G for which p a {p^{a}} is the exact p-part of their order is at least p a - 1 {p^{a-1}} .