Groups with the same order and degree pattern |
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Authors: | Roya Kogani-Moghaddam Ali Reza Moghaddamfar |
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Institution: | 1. Department of Mathematics, K. N. Toosi University of Technology, Tehran, P. O. Box 16315-1618, Iran 2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, P. O. Box 19395-5746, Iran
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Abstract: | The degree pattern of a finite group has been introduced in 18].A group M is called k-fold OD- characterizable if there exist exactly k non-isomorphic finite groups having the same order and degree pattern as M .In particular,a 1-fold OD-characterizable group is simply called OD-characterizable.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 OD-characterizable,while their automorphism groups are 3-fold OD-characterizable.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 3-fold OD-characterizable.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 OD-characterizable,while the symmetric groups S m are OD- characterizable or 3-fold OD-characterizable;(b) the alternating group A 10 is 2-fold OD-characterizable;(c) the symmetric group S 10 is 8-fold OD-characterizable.This theorem completes the study of OD-characterizability of the alternating and symmetric groups A m and S m of degree m 100. |
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Keywords: | OD-characterization of a finite group degree pattern prime graph alternating and symmetric groups |
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