Comment by Geoff Robinson on Generalisation of abelianisation using...
What Isaacs and Passman prove is a little stronger. They show that $G$ has an Abelian normal subgroup $A$ such that $[G:A]$ is bounded in terms of $M$ alone if all complex irreducible characters of $G$...
View ArticleComment by Geoff Robinson on The mysterious significance of local subgroups...
If you think of infinite groups, then there are some groups, such as Tarski monsters which have no non-trivial subgroup structure at all, and you can't do much other than prove they exist.
View ArticleAnswer by Geoff Robinson for Results from abstract algebra which look wrong...
I suppose there is a case for saying that Jordan's theorem on finite complex linear groups might be such a result: there is a function $f: \mathbb{N} \to \mathbb{N}$ such that for every $n \in...
View ArticleAnswer by Geoff Robinson for Finding a primitive idempotent for an...
I am not sure this is worth making an answer, rather than a comment, but in the complex case, we may write the primitive idempotent $e$ in the form$e = \frac{e_{\chi}}{\chi(1)} + \sum_{g \in G}...
View ArticleAnswer by Geoff Robinson for Checking for a normal p-complement with a computer
For the first question, I wonder if either of the following theoretical observations could be made into a probabilistic algorithm: the group $G$ has a normal $p$-complement if and only if $xy$ has...
View ArticleAnswer by Geoff Robinson for Glauberman-Thompson normal $p$-complement...
As regards the last question (if you are asking what I think you are asking), the proof that (for $p$ odd), $G$ has a normal $p$-complement if and only if $N_{G}(ZJ(P))$ has a normal $p$-complement (...
View ArticleAnswer by Geoff Robinson for Group of matrices in which every matrix is...
It does not seem to have been mentioned in your question, or in the answer and comments that a periodic subgroup $G$ of ${\rm GL}(n,\mathbb{C})$ satisfies your condition. There were several theorems of...
View ArticleAnswer by Geoff Robinson for The property of conjugate subgroup
Richard Lyons gave a proof of a more general statement, but I mention an approach to the $p$-group case asked in the question which is (I think) essentially the way Sylow proved (the existence part of)...
View ArticleAnswer by Geoff Robinson for is the embedding $\mathrm{Sp}_{2m}(p)\leqslant...
$\DeclareMathOperator\Sp{Sp}$Derek's answer is definitive. I think, at least for $p> 3$ odd, we can also see that the answer is no by considering the irreducible characters of $\Sp(2m,p)$. The...
View ArticleAnswer by Geoff Robinson for Splitting field for $\mathrm{GL}(2,p)$ -...
I realised that my previous version of this answer was irrelevant to the question (though what was said was accurate, as far as it went).I think you can make the desired conclusion if you resort to...
View ArticleAnswer by Geoff Robinson for Are all indecomposable $\mathbb{Z}_+$-modules...
I do not think this is true. Let $G = A_{5}$, let $A$ denote the character ring of $G$. Let $M$ denote the $\mathbb{C}$-span of the Brauer characters of $G$ for the prime $2$, (extended to take the...
View ArticleAnswer by Geoff Robinson for How often does algebraic-conjugacy imply conjugacy?
It is reasonably standard to call a finite group $G$ a rational group if all its complex irreducible characters are rational-valued ( equivalently, if $g \in G,$ then $g$ is conjugate within $G$ to all...
View ArticleAnswer by Geoff Robinson for Sparsity of q in groups PSL(2,q) that are...
Let me expand my earlier comment to a partial answer of "why sparsity?". It is impossible for $|{\rm PSL}(2,p^{m})|$ to have four or fewer different prime factors when$p$ is an odd prime which is...
View ArticleAnswer by Geoff Robinson for Conjugacy of nilpotent injectors in soluble groups
I take a nilpotent injector of a finite solvable group $G$ to be a maximal nilpotent subgroup $M$ of $G$such that $M \cap N$ contains a maximal nilpotent subgroup of $N$ of whenever $N$ is subnormal in...
View ArticleAnswer by Geoff Robinson for A question related to Jordan's theorem on...
I think that if the given representation $G$ is primitive (that is, not only irreducible, but also not induced from any lower dimensional representation), then $f(n) = 6^{n-1}$ will do, by a theorem of...
View ArticleAnswer by Geoff Robinson for Prove that the ideal of $\mathbb{C}G$ generated...
Another way to say it is that, if we label so that the character $\chi$ is associated to the primitive idempotent $e$ of the group algebra, then $e$ does not appear in the expression of any $p_{i}$ as...
View ArticleAnswer by Geoff Robinson for Conjugation by elements of subgroups
Looking more deeply at this question, there is quite a long history of related questions, going back long prior to the classification of finite simple groups, for examples in results of B.Fischer which...
View ArticleAnswer by Geoff Robinson for Is applying Feit–Thompson’s theorem for the...
I do doubt that Feit–Thompson explicitly dealt with that particular group in their Pacific Journal of Mathematics paper. But from fairly early in their paper, one can see that in a group $G$ of the...
View ArticleAnswer by Geoff Robinson for Finite groups with only one $p$-block
This is really a supplement to @DaveBenson's answer, but M.E. Harris, in Theorem 1 of his 1984 Journal of Algebra paper "On the $p$-deficiency class of a finite group", proved a rather precise theorem...
View ArticleAnswer by Geoff Robinson for When are these irreducible complex...
As noted in comments, all irreducible characters of Weyl groups are rational, and, in particular, real.
View ArticleAnswer by Geoff Robinson for Proof of CFSG assuming every simple group is...
I don't know who the "guru" being alluded to might have been. One obvious consequence of every finite non-Abelian simple group being two-generated is that for each finite simple group $G$, the outer...
View ArticleAnswer by Geoff Robinson for Finite groups with bounded centralizers
There is a paper by Daniel Palacin ("Finite groups contain large centralizers", Israel Journal of Mathematics, 244,(2), (2021), 621-624) which proves (without the classification of finite simple...
View ArticleComment by Geoff Robinson on What are the Schur indices of irreducible...
I guess I should have mentioned the Brauer-Speiser theorem, which links the Schur index of real-valued irreducible characters with their Frobenius-Schur indicator.
View ArticleAnswer by Geoff Robinson for Regular orbits for automorphisms of finite...
If I am reading your question correctly, then I think $A_{5}$ is an example where this fails. The automorphism group is isomorphic to $S_{5}$. The only elements of composite order in the automorphism...
View ArticleAnswer by Geoff Robinson for Embedding $\mathrm{SL}_n(3)$ into...
$\DeclareMathOperator\SL{SL}$The idea I had in mind originally when I made the comment was more simple-minded. I'll deal with the case that $n$ is even for ease of exposition.Each $3$-subgroup $S$ of...
View ArticleComment by Geoff Robinson on Number of conjugacy classes of pairs of...
@DaveBenson : To be fair, I had remembered the $p^{\frac{3}{2}}$ question from reading the question at an earlier time, and had just realised how to show that it was over-optimistic. I had set about...
View ArticleComment by Geoff Robinson on Finite-maximal subgroups of orthogonal groups
@YCor : Thanks, I had forgotten that question/answer.
View ArticleComment by Geoff Robinson on Estimating the cardinality of the set of...
You probably can't do better than $n^{c\log{n}}$ for some constant $c$, as an elementary Abelian $2$-group of order $n = 2^{r}$ illustrates. I think I have answered questions like this before, but I...
View ArticleComment by Geoff Robinson on What would you do if you improve your own result...
These days, within the editorial.refereeing process, you often get several opportunites to revise your paper (unless it is rejected outright, in which case your problem takes care of itself, and you...
View ArticleComment by Geoff Robinson on What is known about $\operatorname{gap}(A) =...
Notice that a nilpotent operator $A$ has $r(A) = 0,$ whereas $\|A\|$ can be arbitrarily large, so I wonder what sort of answer you would regard as useful in your context?
View ArticleComment by Geoff Robinson on Number of conjugacy classes of pairs of...
Since I realise that $G = {\rm PSL}(2,7)$ also attains the bound $5p-3$ with $p = 7,$ I am inclined to think that it would be a delicate matter to actually prove the $5p-3$ bound in the non-solvable case.
View ArticleComment by Geoff Robinson on Is there a way to find the eigenvalues of a...
If you are decomposing a character in the way you seem to be, you are not just dealing with a single matrix (unless you know about group determinants), you are dealing with all the matrices in a...
View ArticleComment by Geoff Robinson on The number of irreducible characters of simple...
Have now put in a link, but you may need other means of access
View ArticleAnswer by Geoff Robinson for Holt's Theorem on doubly transitive groups with...
This may be hard to do without CFSG. For example, if $G$ acts doubly transitively on one of its conjugacy classes of elements of order $p$, this is comparable in difficulty (and a special case of)...
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