on the total character of finite groups
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abstract
for a finite group $g$, we study the total character $tau_g$ afforded by the direct sum of all the non-isomorphic irreducible complex representations of $g$. we resolve for several classes of groups (the camina $p$-groups, the generalized camina $p$-groups, the groups which admit $(g,z(g))$ as a generalized camina pair), the problem of existence of a polynomial $f(x) in mathbb{q}[x]$ such that $f(chi) = tau_g$ for some irreducible character $chi$ of $g$. as a consequence, we completely determine the $p$-groups of order at most $p^5$ (with $p$ odd) which admit such a polynomial. we deduce the characterization that these are the groups $g$ for which $z(g)$ is cyclic and $(g,z(g))$ is a generalized camina pair and, we conjecture that this holds good for $p$-groups of any order.
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Journal title:
international journal of group theoryPublisher: university of isfahan
ISSN 2251-7650
volume 3
issue 3 2014
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