Let (G, Z(G)) be a Camina pair. We prove that ∣G: Z(G)∣ > ∣Z(G)∣² when G has nilpotence class at least 4. When G has nilpotence class 3, we show that ∣G: Z(G)∣ > ∣Z(G)∣^3/2. Under the additional assumption that Z₂(G) < C_G(G′), the inequality ∣G: Z(G)∣≥ ∣Z(G)∣²∣G′: Z(G)∣ is established. If Z₂(G) < C_G(G′) and G has nilpotence class at least 4, then the inequality ∣G: Z(G)∣ > ∣Z(G)∣³ is obtained.

中文翻译：

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中央腰带对

令（G，Z（G））为Camina对。我们证明|摹：Z ^（摹）|> | Z ^（摹）| ²时摹具有幂零类至少4.当摹具有幂零3类，我们证明了| g ^：Z ^（摹）|> | Z ^（G）∣ ^3/2。在Z ₂（G）< C _G（G'）的附加假设下，不等式∣ G：Z（G）∣≥ ∣ Z（G）∣ ² ∣ G'：Z（G）∣成立。如果Z ₂（G）< C _G（G'）且G的幂等性等级至少为4，则获得不等式∣ G：Z（G）∣> ∣ Z（G）∣ ³。