GROUPS WHOSE ELEMENTS COMMUTE WITH THEIR ENDOMORPHIC IMAGES

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FAGHIHI AFARANI A.

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Seminar Paper

Paper Information

Seminar: IRANIAN ALGEBRA SEMINAR
Year:2009 | Issue:20

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Title

GROUPS WHOSE ELEMENTS COMMUTE WITH THEIR ENDOMORPHIC IMAGES

Author(s)

FAGHIHI AFARANI A.

Keywords

ENDOMORPHISM OF GROUPS

2-ENGEL GROUPS

P-GROUPS

NEARRINGS

Abstract

A GROUP G IS CALLED AN E-GROUP IF THE NEAR-RING GENERATED BY THE ENDOMORPHISMS OF G IN THE NEAR-RING OF MAPS ON G IS A RING. IT IS WELL KNOWN (SEE, E.G., MALONE, 1995) THAT A GROUP G IS AN E-GROUP IF AND ONLY IF EACH ELEMENT COMMUTES WITH ITS ENDOMORPHIC IMAGES. FOR ANY PRIME NUMBER P, WE CALL AN E-GROUP WHICH IS ALSO A P-GROUP, A PE-GROUP. IN THIS PAPER AT FIRST WE EXPLAIN GENERAL PROPERTIES OF E-GROUPS. ALSO WE PROVE THAT AN INFINITE FINITELY GENERATED E-GROUP IS THE DIRECT PRODUCT OF A CENTRAL TORSION-FREE SUBGROUP AND A FINITE SUBGROUP. NEXT, WE PROVE THAT THERE IS NO 3E-GROUP OF NILPOTENCY CLASS 3 OF ORDER AT MOST 310. ALSO WE CONSTRUCT A GROUP OF CLASS 3 WHICH IS “VERY CLOSE” TO BE AN E-GROUP. THE FOLLOWING QUESTIONS ARE CENTRAL ONES IN THIS PAPER: (1) WHAT IS THE LEAST NUMBER OF GENERATORS OF A FINITELY GENERATED NONABELIAN E-GROUP?(2) WHAT IS THE MINIMUM ORDER OF A FINITE NON-ABELIAN PE-GROUP?WE PROVE THAT THE MINIMAL NUMBER OF GENERATORS OF A FINITELY GENERATED NON-ABELIAN E-GROUP IS 4.IN RESPONSE TO THE QUESTION (2), WE PROVE THAT THE MINIMUM ORDER OF A FINITE NON-ABELIAN PE-GROUP IS P8, FOR ANY ODD PRIME NUMBER P AND THIS ORDER IS 27 FOR P=2.ALSO WE OBTAIN A NEW CLASS OF E-GROUPS.AS WE HAVE FOUND THAT SOME OF OUR RESULTS ARE VALID FOR A VERY LARGER CLASS OF FINITE P-GROUPS THAN PE-GROUPS, WE STUDY A CLASS OF P-GROUPS FOR EVERY PRIME NUMBER P AND WE DENOTE THIS CLASS OF P-GROUPS BY PE. (A FINITE P-GROUP G IS CALLED A PE-GROUP IF G IS A 2-ENGEL GROUP AND ALL ELEMENTS OF ORDER AT MOST PR LIE IN THE CENTER OF G, WHERE PR IS EXPONENT G/G1).WE CLASSIFY ALL 3-GENERATOR PE-GROUPS AND PE-GROUPS WITH CYCLIC DERIVED SUBGROUP AND DETERMINE ENDOMORPHISMS OF 3-GENERATOR PE-GROUPS AND PE-GROUPS. FINALLY WE CLASSIFY ALL PE-GROUPS AND PE-GROUPS OF ORDER AT MOST P7 FOR ANY PRIME NUMBER P.

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Cite

APA: Copy

FAGHIHI AFARANI, A.. (2009). GROUPS WHOSE ELEMENTS COMMUTE WITH THEIR ENDOMORPHIC IMAGES. IRANIAN ALGEBRA SEMINAR. SID. https://sid.ir/paper/904402/en

Vancouver: Copy

FAGHIHI AFARANI A.. GROUPS WHOSE ELEMENTS COMMUTE WITH THEIR ENDOMORPHIC IMAGES. 2009. Available from: https://sid.ir/paper/904402/en

IEEE: Copy

A. FAGHIHI AFARANI, “GROUPS WHOSE ELEMENTS COMMUTE WITH THEIR ENDOMORPHIC IMAGES,” presented at the IRANIAN ALGEBRA SEMINAR. 2009, [Online]. Available: https://sid.ir/paper/904402/en

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