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Finite Groups Isospectral to Simple Groups

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Abstract

The spectrum of a finite group is the set of element orders of this group. The main goal of this paper is to survey results concerning recognition of finite simple groups by spectrum, in particular, to list all finite simple groups for which the recognition problem is solved.

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Acknowledgements

We are grateful to A. S. Staroletov, H. He, R. Shen, A. A. Buturlakin, D. O. Revin, and S. V. Skresanov for their helpful comments and productive suggestions.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Koptyuga 4, Novosibirsk, Russia, 630090

    Maria A. Grechkoseeva, Victor D. Mazurov & Andrey V. Vasil’ev

  2. Department of Mathematics, Chongqing University of Arts and Sciences, Chongqing, 402160, People’s Republic of China

    Wujie Shi

  3. School of Mathematics, Suzhou University, Suzhou, 215006, People’s Republic of China

    Wujie Shi

  4. School of Science, Hainan University, Haikou, 570228, Hainan, People’s Republic of China

    Andrey V. Vasil’ev

  5. School of Science, Jiangnan University, Wuxi, 214122, People’s Republic of China

    Nanying Yang

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  1. Maria A. Grechkoseeva
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  2. Victor D. Mazurov
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  3. Wujie Shi
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  4. Andrey V. Vasil’ev
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Correspondence to Andrey V. Vasil’ev.

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Maria A. Grechkoseeva, Andrey V. Vasil’ev, and Nanying Yang were supported by Foreign Experts program in Jiangsu Province (No. JSB2018014). Andrey V. Vasil’ev was supported by the National Natural Science Foundation of China (No. 12171126). Victor D. Mazurov was supported by the RFBR (No. 20-51-00007). Wujie Shi was supported by the National Natural Science Foundation of China (11171364, 11671063).

Appendix: Simple groups with solved recognition problem

Appendix: Simple groups with solved recognition problem

Table 1 Finite groups isospectral to \(L=L_{n}(q)\), \(q=p^m\), \(d=(n,q-1)\), \(b=((q-1)/d,m)_d\), \(n_0=27\), \(\psi ,\psi _1, \chi \in \langle \varphi \rangle \), \(\eta \in \langle \delta \rangle \), \(|\psi |=(b)_{2'}\), \(|\psi _1|=(m)_3\), \(|\chi |=(b)_2\), \(|\eta |=(d)_2\)
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Table 2 Finite groups isospectral to \(L=U_{n}(q)\), \(q=p^m\), \(d=(n,q+1)\), \(b=(((q+1)/d,m)_d)_{2'}\), \(n_0=27\) \(\psi ,\psi _1, \chi \in \langle \varphi \rangle \), \(|\psi |=b\), \(|\psi _1|=(m)_3\), \(|\chi |=2(m)_2\), \(\gamma =\chi ^{(m)_2}\)
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Table 3 Finite groups isospectral to \(L=S_{2n}(q)\), \(q=p^m\), \(n_0=16\), \(\chi \in \langle \varphi \rangle \), \(|\chi |=(m)_2\)
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Table 4 Finite groups isospectral to \(L=O_{2n+1}(q)\), \(q=p^m\) odd, \(n_0=16\), \(\chi \in \langle \varphi \rangle \), \(|\chi |=(m)_2\)
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Table 5 Finite groups isospectral to \(L=O_{2n}^+(q)\), \(q=p^m\), \(n_0=19\), \(\chi \in \langle \varphi \rangle \), \(|\chi |=(m)_2\)
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Table 6 Finite groups isospectral to \(L=O_{2n}^-(q)\), \(q=p^m\), \(n_0=18\), \(\chi \in \langle \varphi \rangle \), \(|\chi |=2(m)_2\)
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Table 7 Finite groups isospectral to some simple classical groups with disconnected prime graph, \(n\geqslant 5\)
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Table 8 Finite groups isospectral to exceptional groups of Lie type, \(q=p^m\)
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Table 9 Finite groups isospectral to alternating and sporadic groups
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Table 10 Unrecognizable simple groups L and finite groups G such that \(\omega (G)=\omega (L)\) and G is not an almost simple group with socle L
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Grechkoseeva, M.A., Mazurov, V.D., Shi, W. et al. Finite Groups Isospectral to Simple Groups. Commun. Math. Stat. 11, 169–194 (2023). https://doi.org/10.1007/s40304-022-00288-5

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