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Prikladnaya Diskretnaya Matematika, 2017, Number 35, Pages 5–13
DOI: https://doi.org/10.17223/20710410/35/1
(Mi pdm573)
 

This article is cited in 1 scientific paper (total in 1 paper)

Theoretical Foundations of Applied Discrete Mathematics

Homomorphic stability of finite groups

M. I. Kabenyuk

Kemerovo State University, Kemerovo, Russia
Full-text PDF (667 kB) Citations (1)
References:
Abstract: The set $\mathrm{Hom}(G,H)$ of all homomorphisms from a group $G$ to a group $H$ is a group with respect to the operation of pointwise products iff the images of any two such homomorphisms commute element-wise; in this case, the group is commutative. For finite $G$ and $H$, we study algebraic properties of this group and of the union $\mathrm{Im}(G,H)$ of the images of all homomorphisms from $G$ to $H$. Let $\exp(G)$ be the minimal positive integer $n$ such that $x^n=1$ for all $x\in G$, let $G'$ be the commutator subgroup of $G$, $q=\exp(G/G')$, and let $\Omega_q(H)$ be the subgroup of $H$ generated by all elements of order $q$. We obtain the following results.
If $\mathrm{Hom}(G,H)$ is a group, then $\Omega_q(H)$ is commutative and the groups $\mathrm{Hom}(G,H)$ and $\mathrm{Hom}(G/G',\Omega_q(H))$ are isomorphic. Conversely, if $\Omega_q(H)$ is commutative and $\phi(G')=\{1\}$ for all $\phi\in\mathrm{Hom}(G,H)$, then $\mathrm{Hom}(G,H)$ is a group.
If $\mathrm{Im}(G,H)$ is a subgroup of $H$, then it is endomorphically admissible in $H$.
If $G$ is a finite $p$-group such that $\exp(G)=\exp(G/G')=q$ and $H$ is a regular $p$-group, then $\mathrm{Im}(G,H)=\Omega_q(H)$.
Keywords: homomorphism groups, homomorphic stability, finite group, Frobenius group, regular $p$-group.
Bibliographic databases:
Document Type: Article
UDC: 512.542
Language: Russian
Citation: M. I. Kabenyuk, “Homomorphic stability of finite groups”, Prikl. Diskr. Mat., 2017, no. 35, 5–13
Citation in format AMSBIB
\Bibitem{Kab17}
\by M.~I.~Kabenyuk
\paper Homomorphic stability of finite groups
\jour Prikl. Diskr. Mat.
\yr 2017
\issue 35
\pages 5--13
\mathnet{http://mi.mathnet.ru/pdm573}
\crossref{https://doi.org/10.17223/20710410/35/1}
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  • https://www.mathnet.ru/eng/pdm/y2017/i1/p5
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Прикладная дискретная математика
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    References:45
     
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