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I am a former assistant-professor of mathematics at the Univ. of Amsterdam, with specialization in group theory and representation theory of groups. I work for IBM, though not in research.


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revised Normal subgroups in groups of odd order
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comment Normal subgroups in groups of odd order
You got it, Jorge!
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comment Normal subgroups in groups of odd order
@Jorge - yes or again using the lemma!
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revised Normal subgroups in groups of odd order
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revised Normal subgroups in groups of odd order
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comment Normal subgroups in groups of odd order
Well a generalization would be that $|N|=p$ is prime and gcd$(|G|,p-1)=1$.
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comment Normal subgroups in groups of odd order
This is an example of a very good post, where the OP shows his thoughts, analysis and attempts. +1 from me!
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revised Normal subgroups in groups of odd order
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answered Normal subgroups in groups of odd order
Dec
17
comment Is ideal prime or maximal?
No problem - let me get this straight - $\mathbb{Z}[x]/(x-5) \cong \mathbb{Z}$ - prime ideal. And $\mathbb{Z}[x]/(3,x+5) \cong \mathbb{F}_3$ - maximal ideal.
Dec
17
comment Constructing an irreducible representation for a finite group
The $\phi^G$ is the character $\phi$ of $H$ induced to $G$. Since $\phi$ is an irreducible constituent of $\chi_H$, by reciprocity one must find $\chi$ (with the same multiplicity) in the induction of $\phi$ to $G$. So $\chi$ appears in the decomposition of $\phi^G$ exactly $[\chi_H,\phi]=[\chi,\phi^G]$ times. And this number is at least $1$.
Dec
17
revised Is ideal prime or maximal?
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Dec
17
revised Constructing an irreducible representation for a finite group
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Dec
17
revised Constructing an irreducible representation for a finite group
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Dec
17
comment Constructing an irreducible representation for a finite group
Yes, I have the book under my eyes here (first edition) and it says just an "$\mathbb{F}$-irreducible character $\phi$". That must be an irreducible character of $H$. The rest of the exercise also then follows (case $H$ abelian - so $\phi(1)=1$).
Dec
17
revised Constructing an irreducible representation for a finite group
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Dec
17
comment Constructing an irreducible representation for a finite group
"show that there is an irreducible $\mathbb{F}$-character of $G$ with degree at least $\frac{n}{|G:H|}$" ... I think you mean "irreducible $\mathbb{F}$-character of $H$" .. right?
Dec
17
answered Constructing an irreducible representation for a finite group
Dec
17
revised A question in finite group theory with proof using representation theory
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Dec
17
revised A question in finite group theory with proof using representation theory
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