On symmetric words in the symmetric group of degree three

Płonka, Ernest

doi:10.7146/math.scand.a-14997

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2010

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“…In [11], we have proved that S (3) (D 3 ) is an abelian group. Corollary 5.4 together with Corollary 4.3 yields:…”

Section: Symmetric 3-wordsmentioning

confidence: 98%

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Symmetric Words in Dihedral Groups

Płonka

2010

Algebra Colloq.

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Let G be a group and let w = w(x1, x2, . . . , xn) be a word in the absolutely free group Fn on free variables x1, x2, . . . , xn. The set S (n) (G) of all words w such that the equality w(gσ1, gσ2, . . . , gσn) = w(g1, gn, . . . , gn) holds for all g1, g2, . . . , gn ∈ G and all permutations σ ∈ Sn is a subgroup of Fn, called the subgroup of n-symmetric words for G. In this paper, the groups S (2) (Dp) and S (3) (Dp) for dihedral groups Dp are determined, where p > 3 is a prime. In particular, it turns out that the groups S (3) (Dp) are not abelian.The dihedral group D p is a semidirect product of cyclic groups Z 2 and Z p (for a prime p > 2). It can be presented as the set of all pair (ε, i), where ε = ±1 Algebra Colloq. 2010.17:953-962. Downloaded from www.worldscientific.com by THE UNIVERSITY OF OKLAHOMA on 02/03/15. For personal use only.

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“…In [11], we have proved that S (3) (D 3 ) is an abelian group. Corollary 5.4 together with Corollary 4.3 yields:…”

Section: Symmetric 3-wordsmentioning

confidence: 98%

“…It was also proved that the group of symmetric 2-words for the dihedral group D p (p prime) is the direct product of the group D p and the cyclic group Z p . In [11], it was proved that the group of 3-words of the symmetric group S 3 (i.e., the group D 3 ) is abelian. In this paper, we determine the groups S (n) (D p ) for n = 2, 3.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Words in Dihedral Groups

Płonka

2010

Algebra Colloq.

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show abstract

On symmetric words in the symmetric group of degree three

Abstract: A word w(x 1

Cited by 1 publication

References 9 publications

Symmetric Words in Dihedral Groups

Symmetric Words in Dihedral Groups

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