可扩特征标与奇数阶单项群

Normal subgroups of odd-order monomial $p^a q^b$ groups

A finite group $G$ is called monomial if every irreducible character of $G$ is induced from a linear character of some subgroup of $G$. One of the main questions regarding monomial groups is whether or not a normal subgroup $N$ of a monomial group $G$ is itself monomial. In the case that $G$ is a group of even order, it has been proved (Dade, van der Waall) that $N$ need not be monomial. Here we show that, if $G$ is a monomial group of order $p^aq^b$, where $p$ and $q$ are distinct odd primes, then any normal subgroup $N$ of $G$ is also monomial.

On super-monomial characters and groups having two irreducible monomial character degrees

A character of a group is said to be super-monomial if every primitive character inducing it is linear. It is conjectured by Isaacs that every irreducible character of an odd $M$-group is super-monomial. We show that all non linear irreducible characters of lowest degree of an odd $M$-group is super-monomial and provide cases in which one can guarantee that certain irreducible characters of normal subgroups are super-monomial. Finally, we study groups having two irreducible monomial character degrees.

Groups where all the irreducible characters are super-monomial

Isaacs has defined a character to be super monomial if every primitive character inducing it is linear. Isaacs has conjectured that if $G$ is an $M$-group with odd order, then every irreducible character is super monomial. We prove that the conjecture is true if $G$ is an $M$-group of odd order where every irreducible character is a $\{p \}$-lift for some prime $p$. We say that a group where irreducible character is super monomial is a super $M$-group. We use our results to find an example of a super $M$-group that has a subgroup that is not a super $M$-group.

奇数维有限可解群的正交不可约表示

Orthogonal irreducible representations of finite solvable groups in odd dimension

本文证明奇数维正交空间中的有限不可约可解线性群必保持一维正交分解，从而为单项群，并推广了 Rod Gow 的定理。

关于超$M_{p}$-群的一些评论

Some Remarks on Super $M_{p}$-groups

本文研究了超$M_{p}$-群的性质，证明了其奇数阶正规子群为$M_{p}$-群。