Homomorphic stability of finite groups

M. I. Kabenyuk

Prikladnaâ diskretnaâ matematika, no. 1 (2017), pp. 5-13.

Résumé

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 $\mathrm{Im}(G,H)=\Omega_q(H)$.

Détail

Keywords: homomorphism groups, homomorphic stability, finite group, regular $p$-group.
Mots-clés : Frobenius group

Bibliographie

[1] Grinshpon S. Ya., Yeltsova T. A., “Homomorphly stable Abelian groups”, Tomsk State University Journal, 2003, no. 280, 31–33 (in Russian)

[2] Grinshpon S. Ya., Yeltsova T. A., “Homomorphic images of Abelian groups”, Fundam. Prikl. Mat., 14:5 (2008), 67–76 (in Russian) | MR

[3] Fuchs L., Infinite Abelian Groups, v. 1, Academic Press, N.Y.–San Francisco–London, 1970, 289 pp. | MR | MR | Zbl

[4] Shilin I. A., Kityukov V. V., “Homomorphic stability of pairs of small order groups”, Prikladnaya Diskretnaya Matematika, 2011, no. 4(14), 22–27

[5] Shilin I. A., Kityukov V. V., Aleksandrov A. A., “Homomorphism groups computing and homomorphic stability of pairs of finite groups verification”, Prikladnaya Informatika, 37:1 (2012), 111–115 (in Russian) | MR

[6] Brown R., Frobenius Groups and Classical Maximal Orders, Amer. Math. Soc., 2001, 110 pp. | MR | Zbl

[7] Perumal P., On the Theory of the Frobenius Groups, , 2012 http://researchspace.ukzn.ac.za/handle/10413/8853

[8] Hall M., The Theory of Groups, Macmillan, 1959, 434 pp. | MR | Zbl