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What about, finite nonabelian 3-groups of exponent 3? Those are all quotients of the Burnside group $B(m,3)$ for some value for $m$.

This is false. Let $G = E_{16} \rtimes C_2$, where $E_{16}$ denotes the elementary abelian group of order $16$. Then $G$ is special, but $Z(G) \cong C_2 \times C_2$ has index $8$ in $G$.

No; the smallest counterexamples are given by the groups SmallGroup(54,5) and SmallGroup(54,6) (in GAP's SmallGroups library); these are groups of the form $$G_1 = ((3 \times 3) : 3) : 2 \text{ and } …

Does there exist a group $G$ (finite or infinite) with three subgroups $A, B, C \leq G$ satisfying the following three conditions? $A = N_G(A)$, $B = N_G(B)$, $C = N_G(C)$; $AB = BC = CA = G$; $A …

The notion of groups with a BN-pair has been generalized to groups with a root group datum. There is a wonderful paper by Pierre-Emmanuel Caprace and Bertrand Rémy on this topic, which you can downloa …

This is true if the action of the automorphism group is edge-transitive (so in particular the tree has to be biregular). This is precisely the statement of Lemma 2.6(vii and viii) of our paper "Simple …

The dihedral groups of order $2^n$ (with $n \geq 4$) form such a family. Indeed, for such a group, we have $$\operatorname{cs}(G) = \{ 1, 2, 2^{n-2} \} , $$ and they do have the required property that …

Another important example is given by groups acting on graphs, especially (but not only) in finite group theory. Quite a few of the sporadic finite simple groups have actually been discovered as autom …

The former group can be seen as the group of unitriangular $3 \times 3$-matrices over the field with $p$ elements: $$G = \left\{ \begin{pmatrix} 1 & * & * \\ 0 & 1 & * \\ 0 & 0 & 1 \end{pmatrix} \righ …

Your geometry has the property that each of its rank 2 restrictions is a Fano plane. In particular, the type-preserving automorphism group (let's call it $G$) is a subgroup of the automorphism group o …

No. $SL(2,5)$ is a non-simple non-solvable group with the property that all its proper subgroups are solvable.

Yes, $G$ always contains a cyclic subgroup of composite order. Note that all Sylow subgroups of $G$ are cyclic, i.e. $G$ is a Zassenhaus metacyclic group. Such groups have a very precise structure: th …

I am looking for an explicit description of the unipotent radical of a minimal parabolic subgroup of a unitary group, i.e. the group of isometries of a hermitian form, over an arbitrary field. In his …

My answer is definitely less complete than that of Mark Sapir, but in case you want to see an explicit example: the easiest counterexample is the dihedral group $D_{12}$, which you can view either as …

This follows from the fact that retractions are type-preserving, because they are defined using type-preserving isomorphisms between apartments. (See the argument used in the proof of (3.16) in Suzuki …