For a finite group G, subgroup P of G is 2-minimal if B<P, where B=NG(S) for some Sylow 2-subgroup S and B contained in unique maximal P. fields odd characteristic, this paper contains detailed explicit description all the subgroups general orthogonal groups, certain their subgroups.

If a class of finitely generated groups G is closed under isometric amalgamations along free subgroups, then every G ∈ G can be quasi-isometrically embedded in a group Ĝ ∈ G that has no proper subgroups of finite index. Every compact, connected, non-positively curved space X admits an isometric embedding into a compact, connected, non-positively curved space X such that X has no non-trivial fin...

Numerical analysis of Saxifraga in Iran was carried out on the basis of morphological and chorological data. The cluster analysis confirmed that the genus in Iran includes two distinct groups: Group one consists of two subgroups, a) alpine Irano-Turanian endemic species (S. ramsarica, S. iranica, S. wendelboi, S. koelzii), and b) Euxino-Hyrcanian species (S. paniculata subsp. Cartilaginea, S. k...

let $mathfrak{f}$ be a formation and $g$ a finite group. a subgroup $h$ of $g$ is said to be weakly $mathfrak{f}_{s}$-quasinormal in $g$ if $g$ has an $s$-quasinormal subgroup $t$ such that $ht$ is $s$-quasinormal in $g$ and $(hcap t)h_{g}/h_{g}leq z_{mathfrak{f}}(g/h_{g})$, where $z_{mathfrak{f}}(g/h_{g})$ denotes the $mathfrak{f}$-hypercenter of $g/h_{g}$. in this paper, we study the structur...

This paper and its sequel [17] deal with a range of questions about the subgroup structure of infinite symmetric groups. Our concern is with such questions as the following. How can an infinite symmetric group be expressed as the union of a chain of proper subgroups? What are the subgroups that supplement the normal subgroups of an infinite symmetric group? What are the maximal proper subgroups...

1. Morley, Marsh, Baldwin/Lachlan, Zilber; and Macintyre 2 2. The Borovik Program: CFSG and tameness 3 2.1. The setting: tame or not 3 2.2. The basic framework: even, odd, mixed, degenerate 4 3. Mixed type 5 3.1. Strong and weak embedding 5 3.2. Toward weak embedding: U(G) 6 3.3. The subgroups B(G) and D(G) 6 3.4. The strategy 7 3.5. The curious incident of the dog in the night 8 3.6. Weak embe...

We show that many 3-manifold groups have no nonabelian surface subgroups. For example, any link of an isolated complex surface singularity has this property. In fact, we determine the exact class of closed graph-manifolds which have no immersed 1 -injective surface of negative Euler characteristic. We also determine the class of closed graph manifolds which have no nite cover containing an embe...

We study Morse subgroups and boundaries of random right-angled Coxeter groups in the Erdős–Rényi model. show that at densities below $$\left( \sqrt{\frac{1}{2}}-\epsilon \right) \sqrt{\frac{\log {n}}{n}}$$ almost surely have hyperbolic surface subgroups. This implies their contain embedded circles. Further, above \sqrt{\frac{1}{2}}+\epsilon we that, surely, special a group are virtually free. a...