In this paper we prove that any strongly embedded subgroup of a K*-group G of finite Morley rank and odd type that does not interpret any bad field is solvable if its Pruefer 2-rank is at least 2. If the normal 2-rank of G is at least 3 this has two important consequences: If G contains a non-solvable centraliser of an involution, then G does not contain any proper 2-generated core and centralisers of involutions have trivial cores.
A subgroup $H$ of a group $G$ is said to be {\it pronormal} in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for each $g \in G$. Some problems in Finite Group Theory, Combinatorics, and Permutation Group Theory were solved in terms of pronormality, therefore, the question of pronormality of a given subgroup in a given group is of interest. Subgroups of odd index in finite groups satisfy a native necessary condition of pronormality. In this paper we continue investigations on pronormality of subgroups of odd index and consider the pronormality question for subgroups of odd inde…
Let $G$ be a group and $H$ a subgroup of $G$. This note introduces an equivalent definition of hyperbolic embedded subgroup based on Bowditch's approach to relatively hyperbolic groups in terms of fine graphs.
The proper subgroup $B$ of the group $G$ is called {\it strongly embedded}, if $2\inπ(B)$ and $2\notinπ(B \cap B^g)$ for any element $g \in G \setminus B $ and, therefore, $ N_G(X) \leq B$ for any 2-subgroup $ X \leq B $. An element $a$ of a group $G$ is called {\it finite} if for all $ g\in G $ the subgroups $ \langle a, a^g \rangle $ are finite. In the paper, it is proved that the group with finite element of order $4$ and strongly embedded subgroup isomorphic to the Borel subgroup of $U_3(Q)$ over a locally finite field $Q$ of characteristic $2$ is locally finite and isomorphic to the group…
Let $G$ be a group and $H \le K \le G$. We say that $H$ is $c$-embedded in $G$ with respect to $K$ if there is a subgroup $B$ of $G$ such that $G = HB$ and $H \cap B \le Z(K)$. Given a finite group $G$, a prime number $p$ and a Sylow $p$-subgroup $P$ of $G$, we investigate the structure of $G$ under the assumption that $N_G(P)$ is $p$-supersolvable or $p$-nilpotent and that certain cyclic subgroups of $P$ with order $p$ or $4$ are $c$-embedded in $G$ with respect to $P$. New characterizations of $p$-supersolvability and $p$-nilpotence of finite groups will be obtained.
In this paper we investigate the structure of finite $p$-groups with the property that every subgroup of index $p^i$ is powerful for some $i$. For odd primes $p$, we show that under certain conditions these groups must be potent. Then, motivated by a question of Mann, we investigate in detail the case when all maximal subgroups are powerful. We show that for odd $p$ any finite $p$-group $G$ with all maximal subgroups powerful has a regular power structure - with precisely one exceptional case which is a $3$-group of maximal class and order $81$. To show this counterexample is unique we use a c…
We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple (G,X,H) consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H which is quasi-isometrically embedded by the action. Examples include strongly quasi-convex subgroups of relatively hyperbolic groups, convex cocompact subgroups of mapping class groups, many known convex cocompact subgroups of Out(Fn), and groups generated by powers of independent loxodromic WPD elements of a group acting on a Gromov hyperbolic space. We initiate the…
We construct the first examples of finitely presented groups with cubic Dehn function containing a finitely generated infinite torsion subgroup. Moreover, we show that any infinite free Burnside group with sufficiently large odd exponent can be embedded as a subgroup of a finitely presented group with cubic Dehn function.
We determine, for $p$ odd, all saturated fusion systems on a Sylow $p$-subgroup $S$ of the unitary group $SU_4(p)$ and we prove that they are all realizable by finite groups. In particular, we prove that $S$ does not support any exotic fusion systems.
A group $G$ is said to be cut if, for every $g \in G$, each generator of $< \! g \! >$ is conjugated to either $g$ or $g^{-1}$. It is conjectured that a Sylow 3-subgroup $P$ of a cut group $G$ is cut. We prove that this is true if $|G|$ is odd.
In this note, we show that among finite nilpotent groups of a given order or finite groups of a given odd order, the cyclic group of that order has the minimum number of edges in its cyclic subgroup graph. We also conjecture that this holds for arbitrary finite groups.
