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katie
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Dec 3 '14 at 19:26
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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
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