## Subgroups of groups of central type

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- by Kathleen M. Timmer PDF
- Trans. Amer. Math. Soc.
**189**(1974), 133-161 Request permission

## Abstract:

Let $\lambda$ be a linear character on the center*Z*of a finite group

*Z*of a finite group

*H*, such that (1) ${\lambda ^H} = \sum \nolimits _{i = 1}^p {{\phi _i}(1){\phi _i}}$ where the ${\phi _i}$’s are inequivalent irreducible characters on

*H*of the same degree, and (2) if $\sum \nolimits _{i = 1}^p {{m_i}{\phi _i}(x) = 0}$ for some $x \in H$ and nonnegative integers ${m_i}$, then either ${\phi _i}(x) = 0$ for all

*i*or ${m_i} = {m_j}$ for all

*i, j*. The object of the paper is to describe finite groups which satisfy conditions (1) and (2) in terms of the multiplication of the group. If

*S*is a

*p*Sylow subgroup of the group

*H*, and $R = S \cdot Z$, then

*H*satisfies conditions (1) and (2) if and only if (a) $\{ x \in H:{x^{ - 1}}{h^{ - 1}}xh \in Z \Rightarrow \lambda ({x^{ - 1}}{h^{ - 1}}xh) = 1,h \in H\} /Z$ consists of elements of order a power of

*p*in $H/Z$, and these elements form

*p*conjugacy classes of $H/Z$, and (b) the elements of $\{ x \in R:{x^{ - 1}}{r^{ - 1}}xr \in Z \Rightarrow \lambda ({x^{ - 1}}{r^{ - 1}}xr) = 1,r \in R\} /Z$ form

*p*conjugacy classes of $R/Z$.

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## Additional Information

- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**189**(1974), 133-161 - MSC: Primary 20C15
- DOI: https://doi.org/10.1090/S0002-9947-1974-0357574-8
- MathSciNet review: 0357574