Comment by Geoff Robinson on Finite groups with only one $p$-block
M. Harris proved the result of the Mathieu groups you mentioned ( that they were the only simple examples for $p=2$). Brauer may have known the first result you mention, but the $p$-solvable case is...
View ArticleComment by Geoff Robinson on Explicit character tables of non-existent finite...
I think there were examples where character tables of supposed simple groups were known before the group was known to exist.
View ArticleComment by Geoff Robinson on Explicit character tables of non-existent finite...
@TimothyChow : We now know that for any give order, there are at most two no n-isomorphic simple groups of that order, and with that knowledge (and knowing all cases for which it occurs), it should be...
View ArticleComment by Geoff Robinson on Commutator of a group element on a vector space
Among group theorists, your interpretation is definitely the one used, and $\pi(g)v-v$ may be interpreted as a commutator for the reason given by @RichardLyons . Note also that $g$ acts as the identity...
View ArticleComment by Geoff Robinson on Almost simple groups and their involutions...
The first question was studied for many decades by many people, and not well understood without CFSG. As for the second question, the question of classifying transitive permutation groups in which...
View ArticleComment by Geoff Robinson on When are these irreducible complex...
Am I missing some point? I thought that all Weyl groups were rational, so, in particular, real-valued ( see the answers to the question MR134581, for example). Note that the example of $A_{n}$ is...
View ArticleComment 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...
There are unresolved mysteries around this question. You might also mention the so-called Thompson order formula, which proves that if the finite group $G$ has more than one conjugacy class of...
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 ArticleComment by Geoff Robinson on Group generated by two irrational plane rotations
You might be interested in a paper by Radin and Sadun about certain subgroups of ${\rm SO}_{3}(\mathbb{R})$ generated by certain pairs of rotations. The groups they obtain tend to be amalgams of two...
View ArticleComment by Geoff Robinson on Existence of a finitely presented simple group...
Do you have any theoretical reason to believe there might? Or do you have a reason why you would like to see such a group to highlight some particular property?
View ArticleAnswer by Geoff Robinson for Finite groups with integral character table
It was proved by W. Feit and G. Seitz that there are (at most) five non-Abelian simple groups which are not alternating and may occurs as composition factors of groups with integral character table (to...
View ArticleAnswer by Geoff Robinson for Prime divisors of nonabelian simple group and of...
Since outer automorphism groups of finite simple groups of Lie type are rather small solvable groups, and outer automorphisms are products of graph, field, and diagonal automorphisms in general, it...
View ArticleAnswer by Geoff Robinson for Do $F$-traces of simple modules at $p'$-classes...
It is still the case that the $\mathbb{F}_{q}$-valued trace functions of the (say) $\ell$ non-isomorphic simple $\mathbb{F}_{q}$-modules $V_{1},V_{2}, \ldots V_{\ell}$, are linearly independent, where...
View ArticleAnswer by Geoff Robinson for Why descend a representation (of a finite group)...
One interesting (in my opinion)application is due to W. Feit, who used the Schur index (combined with Brauer's characterization of characters) to give a proof that every complex irreducible...
View ArticleAnswer by Geoff Robinson for Has anyone seen this construction of the Weil...
I will attempt to write up a representation-theoretic argument influenced by papers I have seen (and sometimes written) over the years, making use of automorphisms of extraspecial groups. It may come...
View ArticleAnswer by Geoff Robinson for Collecting proofs that finite multiplicative...
The multiplicative group of a finite field $F$ with $n$ elements has at most $d$ solutions of $x^{d} = 1$ for each divisor $d$ of $n.$ It is proved ( or maybe is an exercise) in Herstein's "Topics in...
View ArticleAnswer by Geoff Robinson for Sylow theorems for infinite groups
Amalgams of finite groups provide another example. Let $A$ and $B$ be finite groups and let $C = A \cap B.$ Suppose that $P$ is a Sylow $p$-subgroup of $A$, and that $C$ contains a Sylow $p$-subgroup...
View ArticleAnswer by Geoff Robinson for Fiction books about mathematicians?
"The Housekeeper and the Professor" gives an interesting insight into the though processes of aMathematician, written from the perspective of a non-Mathematician. The author sem to me to have a good...
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...
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