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Groups with the same order and degree pattern

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To the memory of Maria Silvia Lucido

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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Authors and Affiliations

  1. Department of Mathematics, K. N. Toosi University of Technology, Tehran, P. O. Box 16315-1618, Iran

    Roya Kogani-Moghaddam & Ali Reza Moghaddamfar

  2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, P. O. Box 19395-5746, Iran

    Ali Reza Moghaddamfar

Authors
  1. Roya Kogani-Moghaddam
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  2. Ali Reza Moghaddamfar
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Correspondence to Ali Reza Moghaddamfar.

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Kogani-Moghaddam, R., Moghaddamfar, A.R. Groups with the same order and degree pattern. Sci. China Math. 55, 701–720 (2012). https://doi.org/10.1007/s11425-011-4314-6

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