Endomorphism kernel property for finite groups

Heghine Ghumashyan; Jaroslav Guričan

Mathematica Bohemica (2022)

  • Volume: 147, Issue: 3, page 347-358
  • ISSN: 0862-7959

Abstract

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A group G has the endomorphism kernel property (EKP) if every congruence relation θ on G is the kernel of an endomorphism on G . In this note we show that all finite abelian groups have EKP and we show infinite series of finite non-abelian groups which have EKP.

How to cite

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Ghumashyan, Heghine, and Guričan, Jaroslav. "Endomorphism kernel property for finite groups." Mathematica Bohemica 147.3 (2022): 347-358. <http://eudml.org/doc/298469>.

@article{Ghumashyan2022,
abstract = {A group $G$ has the endomorphism kernel property (EKP) if every congruence relation $\theta $ on $G$ is the kernel of an endomorphism on $G$. In this note we show that all finite abelian groups have EKP and we show infinite series of finite non-abelian groups which have EKP.},
author = {Ghumashyan, Heghine, Guričan, Jaroslav},
journal = {Mathematica Bohemica},
keywords = {endomorphism kernel property; nilpotent group; $p$-group},
language = {eng},
number = {3},
pages = {347-358},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Endomorphism kernel property for finite groups},
url = {http://eudml.org/doc/298469},
volume = {147},
year = {2022},
}

TY - JOUR
AU - Ghumashyan, Heghine
AU - Guričan, Jaroslav
TI - Endomorphism kernel property for finite groups
JO - Mathematica Bohemica
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 147
IS - 3
SP - 347
EP - 358
AB - A group $G$ has the endomorphism kernel property (EKP) if every congruence relation $\theta $ on $G$ is the kernel of an endomorphism on $G$. In this note we show that all finite abelian groups have EKP and we show infinite series of finite non-abelian groups which have EKP.
LA - eng
KW - endomorphism kernel property; nilpotent group; $p$-group
UR - http://eudml.org/doc/298469
ER -

References

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  1. Blyth, T. S., Fang, J., Silva, H. J., 10.1081/agb-120037216, Commun. Algebra 32 (2004), 2225-2242. (2004) Zbl1060.06018MR2100466DOI10.1081/agb-120037216
  2. Blyth, T. S., Fang, J., Wang, L.-B., The strong endomorphism kernel property in distributive double p -algebras, Sci. Math. Jpn. 76 (2013), 227-234. (2013) Zbl1320.06009MR3330070
  3. Blyth, T. S., Silva, H. J., 10.1080/00927870801937240, Commun. Algebra 36 (2008), 1682-1694. (2008) Zbl1148.06005MR2424259DOI10.1080/00927870801937240
  4. Fang, J., 10.1007/s11225-017-9722-3, Stud. Log. 105 (2017), 995-1013. (2017) Zbl1421.06003MR3704306DOI10.1007/s11225-017-9722-3
  5. Fang, G., Fang, J., The strong endomorphism kernel property in distributive p -algebras, Southeast Asian Bull. Math. 37 (2013), 491-497. (2013) Zbl1299.06017MR3134913
  6. Fang, J., Sun, Z.-J., 10.1007/s00012-013-0254-z, Algebra Univers. 70 (2013), 393-401. (2013) Zbl1305.06004MR3127981DOI10.1007/s00012-013-0254-z
  7. Fang, J., Sun, Z.-J., 10.1007/s10114-020-9444-8, Acta Math. Sin., Engl. Ser. 36 (2020), 1076-1082. (2020) Zbl07343749MR4145699DOI10.1007/s10114-020-9444-8
  8. Gaitán, H., Cortés, Y. J., The endomorphism kernel property in finite Stone algebras, JP J. Algebra Number Theory Appl. 14 (2009), 51-64. (2009) Zbl1191.06007MR2548439
  9. Group, GAP, GAP - Groups, Algorithms, and Programming, Version 4.10.2, Available at https://www.gap-system.org/. 
  10. Guričan, J., The endomorphism kernel property for modular p -algebras and Stone lattices of order n , JP J. Algebra Number Theory Appl. 25 (2012), 69-90. (2012) Zbl1258.06002MR2976467
  11. Guričan, J., A note on the endomorphism kernel property, JP J. Algebra Number Theory Appl. 33 (2014), 133-139. (2014) Zbl1302.08004
  12. Guričan, J., 10.17654/JPANTAJun2015_241_258, JP J. Algebra Number Theory Appl. 36 (2015), 241-258. (2015) Zbl1333.06025DOI10.17654/JPANTAJun2015_241_258
  13. Guričan, J., Ploščica, M., 10.1007/s00012-016-0370-7, Algebra Univers. 75 (2016), 243-255. (2016) Zbl1348.06008MR3515400DOI10.1007/s00012-016-0370-7
  14. Halušková, E., 10.21136/mb.2017.0056-16, Math. Bohem. 143 (2018), 161-171. (2018) Zbl06890412MR3831484DOI10.21136/mb.2017.0056-16
  15. Halušková, E., 10.21136/MB.2019.0128-18, Math. Bohem. 145 (2020), 401-414. (2020) Zbl07286021MR4221842DOI10.21136/MB.2019.0128-18
  16. Kaarli, K., Pixley, A. F., 10.1201/9781482285758, Chapman & Hall/CRC, Boca Raton (2001). (2001) Zbl0964.08001MR1888967DOI10.1201/9781482285758
  17. Kurzweil, H., Stellmacher, B., 10.1007/b97433, Universitext. Springer, New York (2004). (2004) Zbl1047.20011MR2014408DOI10.1007/b97433

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