23
votes
Explicit character tables of non-existent finite simple groups
This is not really a proper answer, but it's way too long for a comment:
My understanding is that by the time a complete character table has been obtained, this is very strong evidence for the ...
- 10.3k
14
votes
The mysterious significance of local subgroups in finite group theory
There is indeed a strong analogy between the study of $p$-local subgroups and the theory of buildings, at least for groups of Lie type.
More precisely, if $G$ is a finite group of Lie type over a ...
- 6,454
7
votes
Accepted
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 ...
- 41.8k
7
votes
Finite groups with only one $p$-block
If $G$ has a normal $p$-subgroup containing its centraliser then it only has one $p$-block. This goes back to Brauer.
If $G$ is simple and $p$ is odd, then $G$ has more than one $p$-block (Brockhaus ...
- 10.3k
7
votes
Finite 2-groups with $(ab)^{2}=(ba)^{2}$
The first observation is that the given condition is equivalent to all squares being central, because substituting $c=ab$ in the relation turns it into $c^2=bc^2b^{-1}$. Next, to explain Derek Holt's ...
- 10.3k
7
votes
Accepted
Minimal irrep of $\mathrm{SL}(2,2^r) $
If $q=2^n$ then $q+1$ is divisible by $3$ if and only if $n$ is odd. This is precisely when, on the cyclic subgroup of order $q+1$ there is a non-trivial one dimensional character $\chi$ for which $\...
- 10.3k
6
votes
The mysterious significance of local subgroups in finite group theory
I think this question is vast, and that there is no single answer.
My first remark might be that to produce ANY proper non-cyclic subgroups of a finite group $G$, we have to allow $G$ so act on some ...
- 41.8k
6
votes
Where can I find a table of the exponents of the sporadic groups?
I couldn't find an online table of exponents for sporadic groups, so I used GAP to produce one:
$$
\begin{align*}
\mathbf{Group}&&\mathbf{Exponent}&&\mathbf{Factorization}\\
M_{11}&...
- 161
3
votes
Accepted
Maximal abelian subgroups of an extraspecial group of order $2^{2m+1}$
Your question seems to refer to part (b) of Satz 13.8 on pages 355 and 356.
Reading the text carefully there, the idea is that you take all $a\in U$ and split each into $a_1\cdot a_2$. Then you define ...
- 5,082
3
votes
Group homology for a metacyclic group
Everything follows from the p-primary decomposition theorem, which is nicely explained in Ken Brown's group cohomology bible (Kasper's answer is the restriction-corestriction argument written circa ...
- 16.9k
2
votes
Finitely generated G, such that x^3 = 1 for all x, is finite?
For a (partial) proof:
First, check in some way that for any 4 elements in such a group we have $[x,[y,[z,t]]]=1$. I.e., in the free group, this commutator $[x,[y,[z,t]]]$ is a product of cubes. Thus,...
- 59.4k
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