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Apr 30 |
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element of order n such that $\pi(n)=\pi(G)$, where $\pi(n)$ denote the prime divisors of $n$ Perfectness doesn't help: Just take the direct product of $k$ copies of $G$ where $k$ is the number of prime divisors of $G$. |
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Apr 15 |
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References on techniques for solving equations with discontinuous functions such as floor and ceiling? Your special equation you can first solve $\bmod c$, and then use a solution $s\in \mathbb{Z}$ to write $a = s+a'$ where $a'$ is a multiple of $c$. Now solve for $a'$. |
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Apr 5 |
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Centralizers in non-abelian groups @S. Carnahan: Oh, right. Thanks. Even the very first comment corrects already the statement that no such groups are known. It seems worth to read the comments sometimes... |
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Apr 5 |
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Centralizers in non-abelian groups Do you ask this because of your interest in the commuting graphs of groups? There seems to be no (finite?) group known to have this property - see [the last sentence here](symomega.wordpress.com/2010/03/02/…). |
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Mar 21 |
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Is there a database somewhere for sharing translations of mathematical works? (Or, is anyone interested in a translation of a letter Weil wrote to de Rham in 1946?) Didn't Gauss publish in Latin? |
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Mar 19 |
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Richness of the subgroup structure of p-groups @StefanKohl: Yes, taking $H$ bigger than $Z_p$ doesn't give any improvements. But starting with $n=2$ and $f_p(2)=3$ you can get per induction with $H = Z_p$ upper bounds $f_p(n)\le 3p^{n-2}+(p^{n-2}-1)/(p-1)$. For odd $p$ I get with $f_p(3)=5$ an upper bound of type $O(p^{n-3})$. |
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Mar 18 |
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Richness of the subgroup structure of p-groups Did you try calculating $f_p(n)$ for small $n$ like $2$, $3$ or $4$? Maybe you can improve François' bound slightly by combining these results with the fact that every extension $G$ of a group $N$ by a group $H$ (i.e., $G/N = H$ for some embedding of $N$ into $G$) is a subgroup of the wreath product of $N$ by $H$. |
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Mar 12 |
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Why should one still teach Riemann integration? added hints how to define the Henstock-Kurzweil integral |
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Dec 13 |
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Exponent of Sylow $p$-subgroup of classical groups over a field of characteristic $p$ "largest power of $p$ greater than or equal to $n$"? |
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Nov 22 |
answered | Fastest way to factor integers < 2^60 |
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Nov 21 |
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How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space? I just found also mathoverflow.net/questions/97105/… which might be of interest for you. |
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Nov 21 |
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How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space? For representations of $H$ over $\mathbb{F}_p$ with $p$ coprime to the order $H$ you get - to my (non-expert) knowledge - only similar behavior as over $\mathbb{C}$ if $\mathbb{F}_p$ contains an $n$-th root for $n$ the exponent of the group $H$ (= least common multiple of the orders of all group elements). See also Geoffrey's answer to mathoverflow.net/questions/91132/… |
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Nov 21 |
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How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space? Do I understand you correctly that you mean with "the entire $p'$-group $H$ acts fixed point freely" that each element of $H^#$ acts fixed point freely, whereas in your question you only ask for no nontrivial fixed point for all the group? [For elementwise fixed point free action you surely get strong restrictions on the structure of $H$ - see for example Theorem 8.3.2 in "The Theory of Finite Groups" by Kurzweil/Stellmacher.] |
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Nov 20 |
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How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space? Do you know anything nontrivial about finite $p'$-groups $H$ acting fixed point freely on elementary abelian $p$-groups? |
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Nov 20 |
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How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space? Maybe too hard? You could have started with the easier question asking which $H$ act fixed point freely on an elementary abelian $p$-group, or -- if you already know the solution to this question -- given your readers some hints about what you know about $H$. |
