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JOURNALS // Diskretnaya Matematika // Archive

Diskr. Mat., 2023 Volume 35, Issue 2, Pages 18–33 (Mi dm1746)

This article is cited in 1 paper

Realization of even permutations of even degree by products of four involutions without fixed points

F. M. Malyshev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: We consider representations of an arbitrary permutation $\pi$ of degree $2n$, $n\geqslant3$, by products of the so-called $(2^n)$-permutations (any cycle of such a permutation has length 2). We show that any even permutation is represented by the product of four $(2^n)$-permutations. Products of three $(2^n)$-permutations cannot represent all even permutations. Any odd permutation is realized (for odd $n$) by a product of five $(2^n)$-permutations.

Keywords: alternating group, permutation, involution, generator, cyclic structure, length of an element of a group.

UDC: 512.542.74 + 512.543.1

Received: 10.07.2022

DOI: 10.4213/dm1746


 English version:
Discrete Mathematics and Applications, 2024, 34:5, 263–276


© Steklov Math. Inst. of RAS, 2025