Geoff Robinson
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Registered User
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10h |
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How much of character theory can be done without Schur’s lemma or the Artin-Wedderburn theorem? Several years ago there was quite a large literature on mathematical enities known as "table algebras". The idea originated with R. Brauer I believe, in an attempt to isolate formal properties of character tables sufficient to characterize them (no pun intended). One person who did a good deal of work in this area was H. Blau. |
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May 18 |
answered | On finite groups with same complex-valued character table |
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May 17 |
revised |
Why are Schur multipliers of finite simple groups so small? mentioned bad bounds for $p$-groups |
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May 17 |
answered | Why are Schur multipliers of finite simple groups so small? |
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May 16 |
answered | A catalog of faithful representations of finite groups? |
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May 15 |
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Reference/quote request: “All of combinatorics is the representation theory of $S_n$” The converse may be true: the representation theory of the symmetric group can be reduced to combinatorics. |
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May 15 |
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Special automorphisms of extraspecial groups I corrected my post- I was thinking about the exponent $p$ case. The other direction is OK, as I explained, because the whole of ${\rm Sp}(2r,p)$ is induced by the action of ${\rm SL}(V)$ since the representation does extend to the appropriate central extension of $H,$ which has a subgroup $Z(G) \times {\rm Sp}(2r,p).$ |
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May 14 |
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Proof of the weak Goldbach Conjecture @H A Helfgott: I was intending to be supportive of you in the sense that given that the result was known to be true in all but a finite number (however ridiculously large) number of cases, there would seem to be no reason not to believe that the result had now been proved. Sorry if it sounded otherwise, that was not my intention at all. |
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May 14 |
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Proof of the weak Goldbach Conjecture Didn't Vinogradov prove it for sufficiently large odd numbers in something like 1937? So, it seems reasonable to believe that deciding the question one way or the other would be a matter of time after that. |
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May 11 |
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a Reference for sylow $p-$subgroup Theorem of GL This is proved in Herstein's book "Topics in Algebra" for example-maybe second or third edition. Once you know the order of ${\rm GL}$,it is a matter of verifying that the index of the subgroup of upper unitriangular matrices is prime to $p,$ which is clear. |
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May 8 |
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Modular reductions of simple characters @F.Ladisch: Thanks, that sounds right. |
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May 8 |
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Modular reductions of simple characters @JIm; I can't remember where it is, and I don't have the book to hand, but I'm pretty sure it's there somewhere. It's not the easiest book to read, I agree, bjut there is a lot in there which is hard to find elsewhere. |
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May 8 |
answered | Modular reductions of simple characters |
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May 5 |
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Odd-order groups with homocyclic sylow subgroups I am not aware of one. Do you think they would be substantially different from odd order groups with all Sylow subgroups Abelian? |
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May 4 |
revised |
Relationship between the number of Sylow subgroups with element orders in finite group deleted 1 characters in body |
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May 4 |
accepted | Relationship between the number of Sylow subgroups with element orders in finite group |
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May 4 |
answered | Relationship between the number of Sylow subgroups with element orders in finite group |
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May 2 |
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Does anyone know where I can get a copy of Gaunce Lewis’s thesis? I was also forbidden to access it |
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May 2 |
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Does anyone know where I can get a copy of Gaunce Lewis’s thesis? Peter May does appear on here occasionally. Can't you contact the math department at Chicago? They must have a copy, though maybe not in electronic form. |
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Apr 29 |
accepted | element of order n such that $\pi(n)=\pi(G)$, where $\pi(n)$ denote the prime divisors of $n$ |
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Apr 29 |
accepted | Why didn’t finite group theorists consider groups where all centralizers of non-identity elements are solvable? |
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Apr 28 |
revised |
element of order n such that $\pi(n)=\pi(G)$, where $\pi(n)$ denote the prime divisors of $n$ typo |
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Apr 28 |
answered | element of order n such that $\pi(n)=\pi(G)$, where $\pi(n)$ denote the prime divisors of $n$ |
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Apr 28 |
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Intertwining number My reading would be that it allows irreducible representations which are non finite dimensional, but intends them always to be irreducible. The statement is not true in general, even for finite groups, if the representations are not absolutely irreducible. |
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Apr 27 |
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Why didn’t finite group theorists consider groups where all centralizers of non-identity elements are solvable? minor amendment |
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Apr 27 |
answered | Why didn’t finite group theorists consider groups where all centralizers of non-identity elements are solvable? |
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Apr 23 |
revised |
About isomorphism of $PGL(2)$ and $SO(3)$ Minor amedment |
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Apr 23 |
accepted | About isomorphism of $PGL(2)$ and $SO(3)$ |
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Apr 23 |
revised |
About isomorphism of $PGL(2)$ and $SO(3)$ minor amendement |
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Apr 23 |
answered | About isomorphism of $PGL(2)$ and $SO(3)$ |
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Apr 22 |
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Upper bound on order of finite subgroups of GL_n(Z_p)? @Jim: Well, I suppose the argument enough relies on realisability over $\mathbb{Q}[\omega],$ which does require Brauer's characterization of characters and Brauer's induction theorem ( though I suppose that realisability over the algebraic closure of the rationals, hence over some finite extension of the rationals for a given finite group would do, and that must have be classically known). |
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Apr 21 |
revised |
Upper bound on order of finite subgroups of GL_n(Z_p)? edited body |
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Apr 21 |
revised |
Upper bound on order of finite subgroups of GL_n(Z_p)? added 16 characters in body; deleted 8 characters in body |
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Apr 21 |
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Upper bound on order of finite subgroups of GL_n(Z_p)? @Pete: This works for a fixed $\mathbb{K}$ or $R.$ But I am not sure that the bound only depends on n and p, as Jim sought. I do not think there is a bound which only depends on n and p, as you can't a priori bound the number of p′-roots of unity in R just in terms of n and p. |
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Apr 21 |
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Upper bound on order of finite subgroups of GL_n(Z_p)? @Jim: I have re-edited my answer to give a bit more detail. |
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Apr 21 |
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Upper bound on order of finite subgroups of GL_n(Z_p)? Expanded some comments; added 2 characters in body |
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Apr 21 |
revised |
Upper bound on order of finite subgroups of GL_n(Z_p)? Filled in some details |
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Apr 20 |
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Upper bound on order of finite subgroups of GL_n(Z_p)? @ayanta: I meant finite, though it is also finite index. I gree with the rest of what you say, but left it unsaid. |
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Apr 20 |
answered | Upper bound on order of finite subgroups of GL_n(Z_p)? |
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Apr 16 |
awarded | ● Yearling |
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Apr 14 |
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Isomorphic maximal commutative semi-simple sub algebras of M_n(C). The answer is yes. Each corresponds to a decomposition of the identity as a sum of mutually orthogonal primitive idempotents, each of which has rank 1. |
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Apr 14 |
accepted | wreath product and matrix presentation |
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Apr 13 |
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wreath product and matrix presentation @jp: But I think you mean a Sylow $2$-subgroup of ${\rm GL}(2^{r-1},q)$) when $q \equiv 3$ (mod 4). |
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Apr 13 |
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wreath product and matrix presentation @jp: Yes, I mean a Sylow $2$-subgroup of $S_{2^{r}}.$ |
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Apr 13 |
answered | wreath product and matrix presentation |
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Apr 13 |
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wreath product and matrix presentation @Derek: Presumably the group intended is the semidirect product oF elementary Abelian $2$-group of rank $2^{r-1}$ with a Sylow $2$-subgroup of the symmetric group $S_{2^{r-1}},$ or alternatively a Sylow $2$-subgroup of the group of all monomial $2^{r-1} \times 2^{r-1}$ matrices whose only non zero entries are $\pm 1.$ |
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Apr 13 |
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wreath product and matrix presentation You should specify which field(s) you want to work over. |
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Apr 9 |
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Finite subgroups of $PGL(3,K)$ To amplify: The Fitting subgroup of your group is cyclic of order $21.$ Take a faithful 1-dimensional representation of that Fitting subgroup. It induces to an absolutely irreducible representation of degree 3. The induced representation is 3-dimensional, and is realized over the field generated by the $21$st-roots of unity, a degree $12$ extension of the rationals. The field generated by the character is actually a degree $4$ extension, generated by a primitive cube root of unity and $\sqrt{-7}.$ |
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Apr 9 |
revised |
Finite subgroups of $PGL(3,K)$ added 11 characters in body |
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Apr 9 |
accepted | Finite subgroups of $PGL(3,K)$ |
