Let G be a finite non-abelian group satisfying Phi(G) = 1 and denote by U the nilpotent residual of G. In this paper, we prove that if G is of odd order then U (U - 1) >= 2[G: Z(G)], and if G is of even order not divisible by a Mersenne or a Fermat prime then U(U - 1) >= [G : Z(G)]. These results are best possible and the assumption Phi(G) = 1 cannot be omitted.
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机译：令G为满足Phi（G）= 1的有限非阿贝尔群，并用U表示G的幂等残差。在本文中，我们证明如果G为奇数阶，则 U （ U -1） > = 2 [G：Z（G）]，并且如果G不能被梅森或费马素数整除，则 U （ U -1）> = [G：Z（G）]。这些结果是最好的，并且不能省略Phi（G）= 1的假设。
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