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Title

GROUPS WHOSE ALL CAYLEY GRAPHS ARE INTEGRAL

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Abstract

 LET G BE A FINITE GROUP AND LET SÍG\ {1} BE A SET SUCH THAT IF A Î S, THEN A-1 Î S, WHERE 1 DENOTES THE IDENTITY ELEMENT OF G. THE CAYLEY GRAPH CAY (G; S) OF G OVER THE SET SIS THE GRAPH WHOSE VERTEX SET IS G AND TWO VERTICES A AND B ARE ADJACENT WHENEVER AB- 1 Î S. THE ADJACENCY SPECTRUM OF A GRAPH IS THE MULTI SET OF ALL EIGENVALUES OF THE ADJACENCY MATRIX OF THE GRAPH. A GRAPH IS CALLED INTEGRAL WHENEVER ALL ADJACENCY SPECTRUM ELEMENTS ARE INTEGER. FOLLOWING KLOTZ AND SANDER, WE CALL A GROUP G CAYLEY INTEGRAL WHENEVER ALL CAYLEY GRAPHS OVER G ARE INTEGRAL. FINITE ABELIAN CAYLEY INTEGRAL GROUPS ARE CLASSIFIED BY KLOTZ AND SANDER AS FINITE ABELIAN GROUPS OF EXPONENT DIVIDING 4 OR 6. KLOTZ AND SANDER HAVE PROPOSED THE DETERMINATION OF ALL NONABELIAN CAYLEY INTEGRAL GROUPS. IN THIS PAPER WE COMPLETE THE CLASSIFICATION OF FINITE CAYLEY INTEGRAL GROUPS BY PROVING THAT FINITE NON-ABELIAN CAYLEY INTEGRAL GROUPS ARE THE SYMMETRIC GROUP S3 OF DEGREE 3, C3 × C4 AND Q8 CN 2 FOR SOME INTEGER N ³ 0, WHERE Q8 IS THE QUATERNION GROUP OF ORDER 8.

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