Related papers: Powerfully solvable and powerfully simple groups

It is proved that for any prime $p$ a finitely generated nilpotent group is conjugacy separable in the class of finite $p$-groups if and only if the torsion subgroup of it is a finite $p$-group and the quotient group by the torsion subgroup…

We give a solution stated in the title to problem 3 of part 1 of the problems listed in the book of Eklof and Mekler [EM],(p.453). There, in pp. 241-242, this is discussed and proved in some cases. The existence of strongly lambda-free ones…

If G is a finitely generated powerful pro-p group satisfying a certain law v=1, and if G can be generated by a normal subset T of finite width which satisfies a positive law, we prove that G is nilpotent. Furthermore, the nilpotency class…

In the paper we characterize the class of finite solvable groups by two-variable identities in a way similar to the characterization of finite nilpotent groups by Engel identities. More precisely, a sequence of words $u_1,...,u_n,... $ is…

We introduce the notion of a bicollapsible 2-complex. This allows us to generalize the hyperbolicity of one-relator groups with torsion to a broader class of groups with presentations whose relators are proper powers. We also prove that…

Suppose that G is a finite group and x in G has prime order p > 3. Then x is contained in the solvable radical of G if (and only if) <x,x^g> is solvable for all g in G. If G is an almost simple group and x in G has prime order p > 3 then…

We give the first examples of nonabelian left-orderable groups such that the conjugacy orbit equivalence relation on its space of orders has infinity orbits, yet it is smooth in the Borel sense. The examples are all nilpotent groups and we…

We characterize the strong metric dimension of the power graph of a finite group. As applications, we compute the strong metric dimension of the power graph of a cyclic group, an abelian group, a dihedral group or a generalized quaternion…

We prove that if a solvable group A acts coprimely on a solvable group G, then A has a "large" orbit in its corresponding action on the set of ordinary complex irreducible characters of G. This extends (at the cost of a weaker bound) a 2005…

The Fitting subgroup of a type-definable group in a simple theory is relatively definable and nilpotent. Moreover, the Fitting subgroup of a supersimple hyperdefinable group has a normal hyperdefinable nilpotent subgroup of bounded index,…

The object of this paper is to examine finite solvable groups whose integral group rings have only trivial central units.

In this paper we introduce the notion of a quasi-powerful $p$-group for odd primes $p$. These are the finite $p$-groups $G$ such that $G/Z(G)$ is powerful in the sense of Lubotzky and Mann. We show that this large family of groups shares…

A set $B$ is a basis for a vector space $V$ if every element of $V$ can be uniquely written as a linear combination of the elements of $B$. There is a similar definition of a basis for a finite group. We show that certain semidirect…

This is the second one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with almost simple unitary groups.

We introduce a formalism of infinite, linearly ordered products in general groups. Using this, we define infinite compositions in certain groups of formal power series such as transseries. We show that such groups can sometimes be…

Considering a finite group $G$, for any element $x\in G$, the solvabilizer of $x$ in $G$ is defined as $Sol_G(x)=\{y \in G : \langle x, y \rangle \text{ is solvable}\}$. In this paper, we introduce $Solv(G)$ as the number of distinct…

We study solvability, nilpotency and splitting property for algebraic supergroups over an arbitrary field $K$ of characteristic $\mathrm{char}\, K \ne 2$. Our first main theorem tells us that an algebraic supergroup $\mathbb{G}$ is solvable…

An irreducible ordinary character of a finite reductive group is called quadratic unipotent if it corresponds under Jordan decomposition to a semisimple element $s$ in a dual group such that $s^2=1$. We prove that there is a bijection…

Every finite non-nilpotent group can be extended by a term operation such that solving equations in the resulting algebra is NP-complete and checking identities is co-NP-complete. This result was firstly proven by Horv\'ath and Szab\'o; the…

We study two complexity notions of groups - a computable Scott sentence and the index set of a group. Finding the exact complexity of one of them usually involves finding the complexity of the other, but this is not the case sometimes. J.…