On sets of elements of the same order in the alternating group $A_n$
Copyright policy )On sets of elements of the same order in the alternating group $A_n$
In the paper under review the following theorems are proved. Theorem A: For \(n>4\), every element of \(A_ n\) can be written as the product of two elements of order 2 in \(A_ n\) if and only if \(n\in \{5,6,10,14\}\). Theorem B: For \(n>2\), every element of \(A_ n\) is the product of two elements of order 3. Theorem A is a strengthening of a result of \textit{J. L. Brenner}, \textit{M. Randall} and \textit{J. Riddell} [Colloq. Math. 32, 39-48 (1974; Zbl 0273.20003)]. Theorem B is a rediscovery of a result of \textit{J. L. Brenner} and \textit{J. Riddell} [Am. Math. Mon. 84, 39-40 (1977; Zbl 0445.05002)]. \textit{J. L. Brenner} and \textit{R. J. Evans} [J. Comb. Theory, Ser. A 45, 196-206 (1987; Zbl 0621.20017)] proved the corresponding result for elements of order 5 if \(n\geq 15\).
Subgroups of symmetric groups, Generators, relations, and presentations of groups, Arithmetic and combinatorial problems involving abstract finite groups, products of elements of order 2, products of elements of order 3
Subgroups of symmetric groups, Generators, relations, and presentations of groups, Arithmetic and combinatorial problems involving abstract finite groups, products of elements of order 2, products of elements of order 3
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