Groups with the same order and degree pattern

Authors: Roya KoganiMoghaddam Ali Reza Moghaddamfar

Publish Date: 2011/12/05

Volume: 55, Issue: 4, Pages: 701-720

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Abstract

The degree pattern of a finite group has been introduced in 18 A group M is called kfold ODcharacterizable if there exist exactly k nonisomorphic finite groups having the same order and degree pattern as M In particular a 1fold ODcharacterizable group is simply called ODcharacterizable It is shown that the alternating groups A m and A m+1 for m = 27 35 51 57 65 77 87 93 and 95 are ODcharacterizable while their automorphism groups are 3fold ODcharacterizable It is also shown that the symmetric groups S m+2 for m = 7 13 19 23 31 37 43 47 53 61 67 73 79 83 89 and 97 are 3fold ODcharacterizable From this the following theorem is derived Let m be a natural number such that m ⩽ 100 Then one of the following holds a if m ≠ 10 then the alternating groups A m are ODcharacterizable while the symmetric groups S m are ODcharacterizable or 3fold ODcharacterizable b the alternating group A 10 is 2fold ODcharacterizable c the symmetric group S 10 is 8fold ODcharacterizable This theorem completes the study of ODcharacterizability of the alternating and symmetric groups A m and S m of degree m ⩽ 100

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