### Abstract

Given a group G, a mapping α : G → G is said to be a semi-endomorphism of G if α(x + y + x) = α(x) + α(y) + α(x) for all x,y ∈ G. It is shown that any non-trivial zero preserving semi-endomorphism of a finite simple group of order greater than two is either an automorphism or an anti-automorphism. Moreover, the semi-endomorphisms of S_{n}, the symmetric group of degree n, n ≥ 4, are described. As an application, it is proved that the semi-endomorphism nearring S(S_{n}) of S_{n}, with n ≥ 3 is equal to E(S_{n}) + M_{c}(S_{n}), where E(S_{n}) is the endomorphism nearring of S_{n}, and M_{C}(S_{n}) is the nearring of constant mappings of S_{n}.

Original language | English |
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Pages (from-to) | 2193-2205 |

Number of pages | 13 |

Journal | Communications in Algebra |

Volume | 27 |

Issue number | 5 |

Publication status | Published - 1999 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Communications in Algebra*,

*27*(5), 2193-2205.

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*Communications in Algebra*, vol. 27, no. 5, pp. 2193-2205.

**On semi-endomorphisms of groups.** / Beidar, K. I.; Fong, Y.; Ke, Wen-Fong; Wu, W. R.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On semi-endomorphisms of groups

AU - Beidar, K. I.

AU - Fong, Y.

AU - Ke, Wen-Fong

AU - Wu, W. R.

PY - 1999

Y1 - 1999

N2 - Given a group G, a mapping α : G → G is said to be a semi-endomorphism of G if α(x + y + x) = α(x) + α(y) + α(x) for all x,y ∈ G. It is shown that any non-trivial zero preserving semi-endomorphism of a finite simple group of order greater than two is either an automorphism or an anti-automorphism. Moreover, the semi-endomorphisms of Sn, the symmetric group of degree n, n ≥ 4, are described. As an application, it is proved that the semi-endomorphism nearring S(Sn) of Sn, with n ≥ 3 is equal to E(Sn) + Mc(Sn), where E(Sn) is the endomorphism nearring of Sn, and MC(Sn) is the nearring of constant mappings of Sn.

AB - Given a group G, a mapping α : G → G is said to be a semi-endomorphism of G if α(x + y + x) = α(x) + α(y) + α(x) for all x,y ∈ G. It is shown that any non-trivial zero preserving semi-endomorphism of a finite simple group of order greater than two is either an automorphism or an anti-automorphism. Moreover, the semi-endomorphisms of Sn, the symmetric group of degree n, n ≥ 4, are described. As an application, it is proved that the semi-endomorphism nearring S(Sn) of Sn, with n ≥ 3 is equal to E(Sn) + Mc(Sn), where E(Sn) is the endomorphism nearring of Sn, and MC(Sn) is the nearring of constant mappings of Sn.

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UR - http://www.scopus.com/inward/citedby.url?scp=0033245495&partnerID=8YFLogxK

M3 - Article

VL - 27

SP - 2193

EP - 2205

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 5

ER -