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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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