2 votes

A Shelah group in ZFC?

This is a partial answer to Problem 2 about the largest number $n$ such that every $n$-Shelah group is finite. The main result of this paper implies that this number $n<36$, at least under the ...
  • 35.6k
2 votes
Accepted

The property of self-normalizing subgroup

Yes. Nilpotency of $T$ is not needed. Assume that $N_G(P_s)>P_s$. Then for some $i<s$, $N_G(P_s)$ has a Sylow $p_i$-subgroup $Q_i\ne 1$. Assume without loss that $p_i$ divides $|H|$. Then $...
1 vote
Accepted

The dimension of a torsion-free $p$-adic analytic group generated by two generators

A pro-$p$ group $G$ has $\textit{lower rank}$ $r$ if $r$ is minimal such that every open subgroup of $G$ contains an open subgroup generated by at most $r$ elements. Lubotzky and Mann showed that the ...
1 vote

Schur multiplier of $\mathrm{SL}(2,\mathbb{Q})$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$For a field $F$ with $|F|>4$ and $|F|\neq9$, the group $H_2(\SL_2(F))$ has a presentation in terms of the symplectic Steinberg symbols; that ...
  • 329
1 vote
Accepted

A group-theoretic lemma in a paper by Ershov and He

Here is @AndyPutman's comment as an answer (so that it can be accepted), made CW to avoid reputation. If @AndyPutman prefers to post the answer, then I will delete this. This only needs the fact ...
1 vote
Accepted

A result of Borel on extensions of arithmetic groups

A proof appears in Section 2.3 of my paper Mapping class groups of manifolds with boundary are of finite type. As pointed out in the comments, it is crucial that $G$ is unipotent.
  • 7,668
1 vote
Accepted

On the number $n_0$ in Shelah's construction of a Jonsson group

The number $n_0$ in Shelah's proof equals $6640$, but can be lowered (by a minor modification of his method) to $36$, see this preprint.
  • 35.6k

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