An Arad and Fisman’s theorem on products of conjugacy classes revisited. (English) Zbl 07609413
The Arad-Fisman theorem [Z. Arad and E. Fisman, Proc. Edinb. Math. Soc., II. Ser. 30, 7–22 (1987; Zbl 0592.20006)] says that, if \(A\) and \(B\) are conjugacy classes of a non-abelian finite simple group, then \(A\cup B\ne AB\ne A^{-1}\cup B\).
The authors generalise this result and prove (in particular) that for any conjugacy classes \(A\) and \(B\) of a finite group,
The authors generalise this result and prove (in particular) that for any conjugacy classes \(A\) and \(B\) of a finite group,
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- if \(AB=A\cup B\), then \(\langle A\rangle=\langle B\rangle\) is solvable;
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- if \(AB=A^{-1}\cup B\) with \(A\ne A^{-1}\), then \(A=B\) and \(\langle A\rangle\) is solvable;
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- if \(AA^{-1}=\{1\}\cup A\cup A^{-1}\), then \(A=AA^{-1}\) is elementary abelian.
Reviewer: Anton A. Klyachko (Moskva)
Software:
GAPReferences:
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| [3] | Arad, Z.; Muzychuk, M., Order evaluation of products of subsets in finite groups and its applications. II, Trans. Am. Math. Soc., 349, 11, 4401-4414 (1997) · Zbl 0895.20022 · doi:10.1090/S0002-9947-97-01866-7 |
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| [6] | Beltrán, A.; Felipe, MJ; Melchor, C., Squares of real conjugacy classes in finite groups, Ann. Mat. Pura Appl., 197, 2, 317-328 (2018) · Zbl 1468.20059 · doi:10.1007/s10231-017-0681-0 |
| [7] | Camina, RD, Applying combinatorial results to products of conjugacy classes, J. Group Theory, 23, 5, 917-923 (2020) · Zbl 1476.20032 · doi:10.1515/jgth-2020-0036 |
| [8] | Guralnick, RM; Navarro, G., Squaring a conjugacy class and cosets of normal subgroups, Proc. Am. Math. Soc., 144, 5, 1939-1945 (2016) · Zbl 1346.20037 · doi:10.1090/proc/12874 |
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