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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Subgroups of groups of central type


Author: Kathleen M. Timmer
Journal: Trans. Amer. Math. Soc. 189 (1974), 133-161
MSC: Primary 20C15
MathSciNet review: 0357574
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0357574-8
PII: S 0002-9947(1974)0357574-8
Keywords: Character, representation, group of central type, character of large degree, projective representation, projective group algebra
Article copyright: © Copyright 1974 American Mathematical Society