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 Jul 2 awarded Curious Jun 9 answered Bound on the order of a finite group generated by elements $a$ and $b$ of order 2 and $n \geq 3$ such that the sum of the images of $a$, $b$ and $b^{-1}$ under any ordinary representation has only rational eigenvalues Jun 7 awarded Critic Jun 7 comment Bound on the order of a finite group generated by elements $a$ and $b$ of order 2 and $n \geq 3$ such that the sum of the images of $a$, $b$ and $b^{-1}$ under any ordinary representation has only rational eigenvalues I don't know what you mean by background and motivation. I was came across this problem when I was studying Q-groups in the book "Structure and Representations of Q-Groups" by Dennis Kletzing. I needed to know, what can happen if we put more restriction. Is that produce an infinite class of groups like Q-groups, or not. This was the most obvious extension I could have think of. Characters turning away many other information about representations. Problem get a lot challenging when we look representation. I know the answer for this, if I was just taking characters rather than representations. Jun 6 comment Bound on the order of a finite group generated by elements $a$ and $b$ of order 2 and $n \geq 3$ such that the sum of the images of $a$, $b$ and $b^{-1}$ under any ordinary representation has only rational eigenvalues Update! This result is valid for all groups of order less than 460 up to now (verified by computers). Jun 6 comment Bound on the order of a finite group generated by elements $a$ and $b$ of order 2 and $n \geq 3$ such that the sum of the images of $a$, $b$ and $b^{-1}$ under any ordinary representation has only rational eigenvalues $D_6, D_8, D_{12}, A_4, S_4, A_4\times \mathbb{Z}_2, D_6\times \mathbb{Z}_4, D_8\times \mathbb{Z}_3$ (where $D_n$ is the dihedral group of order $n$) are the only groups I have found satisfying the conditions stated in the problem. Jun 6 comment Bound on the order of a finite group generated by elements $a$ and $b$ of order 2 and $n \geq 3$ such that the sum of the images of $a$, $b$ and $b^{-1}$ under any ordinary representation has only rational eigenvalues @Stefan Kohl, b can be anything other than an involution (and obviously identity) so I edited again. thanks Jun 6 revised Bound on the order of a finite group generated by elements $a$ and $b$ of order 2 and $n \geq 3$ such that the sum of the images of $a$, $b$ and $b^{-1}$ under any ordinary representation has only rational eigenvalues added 14 characters in body; added 12 characters in body Jun 6 revised Bound on the order of a finite group generated by elements $a$ and $b$ of order 2 and $n \geq 3$ such that the sum of the images of $a$, $b$ and $b^{-1}$ under any ordinary representation has only rational eigenvalues added 7 characters in body Jun 6 asked Bound on the order of a finite group generated by elements $a$ and $b$ of order 2 and $n \geq 3$ such that the sum of the images of $a$, $b$ and $b^{-1}$ under any ordinary representation has only rational eigenvalues Jun 5 asked Eigenvalues of Sum of non-singular matrix and diagonal matrix Oct 25 asked power sums are enough for rationality? Oct 18 asked When does a polynomial split over Q? Oct 14 asked The powers of non-empty subset of a group that generate a subgroup Oct 9 revised is there any bound on the absolute number of algebraic integer in terms of its degree? added 278 characters in body Oct 9 asked is there any bound on the absolute number of algebraic integer in terms of its degree? Sep 17 awarded Supporter Sep 17 awarded Editor Sep 17 comment When does the absolute value of a sum of an integer and an algebraic integer equal an integer? So, what is this telling us regarding Z and |Z|?!! Sep 16 revised When does the absolute value of a sum of an integer and an algebraic integer equal an integer? edited tags