Commutative Schur Rings Over A₅

Schur rings are a type of subring of the group ring that is determined by a partition of the group.In Ref.[6],author study commutative Schur rings over the symmetric group Sn that contain the sum of the transpositions in Sn,by determining the possibilities for the partition of the class of transpositions that such a Schur ring gives.Starting from the definition and nature of the Schur ring,this paper uses mathematical software GAP to determine the commutative Schur rings over A5 that contain the sum of all the 3-cycles in A5.Through calculation and proof,this paper determines that there are eleven such types(up to conjugacy),of which two have the set of all the 3-cycles as a principal set of the Schur ring.Lastly,some commutative Schur rings over A6 and A7 are calculated in order to find some rules to determine the commutative Schur rings over alternating group An.