For a character χ of a finite group G, the number ${\chi }^c(1)=\frac{[G:\ker \chi ]}{\chi (1)}$ is called the codegree of χ. Let N be a normal subgroup of G and set

$$\mathrm{Irr}(G|N)=\mathrm{Irr}(G)-\mathrm{Irr}(G/N).$$

Let p be a prime. In this paper, we first show that if for two distinct prime divisors p and q of $|N|$, $pq$ divides none of the codegrees of elements of $\mathrm{Irr}(G|N)$, then $\mathrm{Fit}(N)\ne \{1\}$ and N is either p‐solvable or q‐solvable. Next, we classify the finite groups with exactly one irreducible character of the codegree divisible by p and, also finite groups whose codegrees of irreducible characters which are divisible by p are equal. Then, we prove that p‐length of a finite p‐solvable group is not greater than the number of the distinct codegrees of its irreducible characters which are divisible by p. Finally, we consider the case when the codegree of every element of $\mathrm{Irr}(G|N)$ is square‐free.

中文翻译：

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拟合子组，p长度，派生长度和字符表

对于有限群G的字符χ ，该数字${\chi }^C（\mathrm{1个}）=\frac{[G：\mathrm{克尔}\chi ]}{\chi （\mathrm{1个}）}$称为χ的codegree。令N为G的一个正常子群并设

$$\mathrm{Irr}（G|ñ）=\mathrm{Irr}（G）-\mathrm{Irr}（G/ñ）。$$

令p为质数。在本文中，我们首先表明，如果两个不同的素因子p和q的$|ñ|$， $pq$ 不除的元素的代码格 $\mathrm{Irr}（G|ñ）$， 然后 $\mathrm{合身}（ñ）\ne \{\mathrm{1个}\}$和Ñ或者是p -solvable或q -solvable。接下来，我们与由codegree整除的正好一个不可约字符分类的有限群p和，也有限群其的不可约特征，其通过整除codegrees p相等。然后，我们证明了一个有限的p可解基团的p长度不大于其不可约性字符被p整除的不同码格的数量。最后，我们考虑以下情况：$\mathrm{Irr}（G|ñ）$ 没有正方形。