A characterization of Schmidt groups by their endomorphism semigroups

Peeter Puusemp

Home/Archive/arXiv

arXiv · math/0210380

A characterization of Schmidt groups by their endomorphism semigroups

Peeter Puusemp

View original source ↗Read source document ↗Export metadata ↓

Abstract

A {\it Schmidt group} is a non-nilpotent finite group in which each proper subgroup is nilpotent. Each Schmidt group G can be described by three parameters p, q and v, where p and q are different primes and v is a natural number, $v\ge 1$. Denote by ${\mathcal{S}}$ the class of all Schmidt groups which have the same parameters p, q and v. It is shown in this paper that the class ${\mathcal{S}}$ can be characterized by the properties of the endomorphism semigroups of the groups of this class. It follows from this characterization that if $G\in {\mathcal{S}}$ and H is another group such that the endomorphism semigroups of G and H are isomorphic, then $H\in {\mathcal{S}}$, too.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Peeter Puusemp. 2002-10-24. A characterization of Schmidt groups by their endomorphism semigroups. https://arxiv.org/abs/math/0210380

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Finiteness conditions on skew braces and solutions of the Yang-Baxter equation

A finite non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation gives rise to a structure skew brace $B(X,r)$ that is a $λ_f$-skew brace, i.e. every element has finitely many $λ$-images, and whose additive group is $FC$. This motivates the study of finiteness conditions on skew braces. We first study the general class of $λ_f$ skew braces and the subclass where the additive group is $FC$, showing that these properties share a resemblance to finite conjugacy, having an analog of the $FC$-center and several analogous structural results. Furthermore, by passing through the structure skew brace of a solution, this property measures whether elements are contained in a finite decomposition factor, identifying a class of infinite solutions that may exhibit similar properties to finite ones. Finally, we show that for a sub skew brace where both groups have finite index, both indices need to coincide and that such a sub skew brace contains a strong left ideal of finite index.

math.GR↗

arXiv2026

Non-uniform exponential growth and the decay of growth rates in growing dimensions

We provide the first example of a finitely presented, and the first example of a simple, group of non-uniform exponential growth. The example is given by Thompson's group $V$. Our methods also show that the infimal exponential growth rates of $\mathrm{Aut}(F_{2^{n+2}})$ and of $\mathrm{EL}_{2^{n+2}}(R)$, for every finitely generated ring $R$, tend to $1$. As an application, we obtain the first example of an acylindrically hyperbolic group, and the first example of a Kazhdan group, of non-uniform exponential growth.

math.GR↗

arXiv2026

Solvable Supplements to Normalizers of Cyclic 2-Subgroups

Amberg and Kazarin proved that a finite group is solvable if the normalizer of every cyclic subgroup of prime power order has a solvable supplement. We substantially relax this hypothesis by requiring it only for cyclic $2$-subgroups. This condition, denoted by $\mathrm{SSN}_2$, sharply restricts the nonabelian composition factors of the group to the family $\PSL_2(q)$, where $q\geq7$ is a prime power satisfying $q\equiv3\pmod4$. Conversely, this family is precisely the nonabelian finite simple groups that satisfy $\mathrm{SSN}_2$. Consequently, a finite group satisfying $\mathrm{SSN}_2$ is solvable if and only if it has no section isomorphic to one of these groups.

math.GR↗