Geodesic
Augmentation quotients for Burnside rings of generalized dihedral groups
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 4, pp. 1165-1175
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Let $H$ be a finite abelian group of odd order, $\mathcal {D}$ be its generalized dihedral group, i.e., the semidirect product of $C_2$ acting on $H$ by inverting elements, where $C_2$ is the cyclic group of order two. Let $\Omega (\mathcal {D})$ be the Burnside ring of $\mathcal {D}$, $\Delta (\mathcal {D})$ be the augmentation ideal of $\Omega (\mathcal {D})$. Denote by $\Delta ^n(\mathcal {D})$ and $Q_n(\mathcal {D})$ the $n$th power of $\Delta (\mathcal {D})$ and the $n$th consecutive quotient group $\Delta ^n(\mathcal {D})/\Delta ^{n+1}(\mathcal {D})$, respectively. This paper provides an explicit $\mathbb {Z}$-basis for $\Delta ^n(\mathcal {D})$ and determines the isomorphism class of $Q_n(\mathcal {D})$ for each positive integer $n$.
Let $H$ be a finite abelian group of odd order, $\mathcal {D}$ be its generalized dihedral group, i.e., the semidirect product of $C_2$ acting on $H$ by inverting elements, where $C_2$ is the cyclic group of order two. Let $\Omega (\mathcal {D})$ be the Burnside ring of $\mathcal {D}$, $\Delta (\mathcal {D})$ be the augmentation ideal of $\Omega (\mathcal {D})$. Denote by $\Delta ^n(\mathcal {D})$ and $Q_n(\mathcal {D})$ the $n$th power of $\Delta (\mathcal {D})$ and the $n$th consecutive quotient group $\Delta ^n(\mathcal {D})/\Delta ^{n+1}(\mathcal {D})$, respectively. This paper provides an explicit $\mathbb {Z}$-basis for $\Delta ^n(\mathcal {D})$ and determines the isomorphism class of $Q_n(\mathcal {D})$ for each positive integer $n$.
DOI : 10.1007/s10587-016-0316-4
Classification : 16S34, 20C05
Keywords: generalized dihedral group; Burnside ring; augmentation ideal; augmentation quotient 
@article{10_1007_s10587_016_0316_4,
     author = {Chang, Shan},
     title = {Augmentation quotients for {Burnside} rings of generalized dihedral groups},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1165--1175},
     year = {2016},
     volume = {66},
     number = {4},
     doi = {10.1007/s10587-016-0316-4},
     mrnumber = {3572929},
     zbl = {06674868},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0316-4/}
}
TY  - JOUR
AU  - Chang, Shan
TI  - Augmentation quotients for Burnside rings of generalized dihedral groups
JO  - Czechoslovak Mathematical Journal
PY  - 2016
SP  - 1165
EP  - 1175
VL  - 66
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0316-4/
DO  - 10.1007/s10587-016-0316-4
LA  - en
ID  - 10_1007_s10587_016_0316_4
ER  - 
%0 Journal Article
%A Chang, Shan
%T Augmentation quotients for Burnside rings of generalized dihedral groups
%J Czechoslovak Mathematical Journal
%D 2016
%P 1165-1175
%V 66
%N 4
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0316-4/
%R 10.1007/s10587-016-0316-4
%G en
%F 10_1007_s10587_016_0316_4
Chang, Shan. Augmentation quotients for Burnside rings of generalized dihedral groups. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 4, pp. 1165-1175. doi: 10.1007/s10587-016-0316-4

[1] Bak, A., Tang, G.: Solution to the presentation problem for powers of the augmentation ideal of torsion free and torsion Abelian groups. Adv. Math. 189 (2004), 1-37. | DOI | MR | Zbl

[2] Chang, S.: Augmentation quotients for complex representation rings of point groups. J. Anhui Univ., Nat. Sci. 38 (2014), 13-19 Chinese. English summary. | MR | Zbl

[3] Chang, S.: Augmentation quotients for complex representation rings of generalized quaternion groups. Chin. Ann. Math., Ser. B. 37 (2016), 571-584. | DOI | MR | Zbl

[4] Chang, S., Chen, H., Tang, G.: Augmentation quotients for complex representation rings of dihedral groups. Front. Math. China 7 (2012), 1-18. | DOI | MR | Zbl

[5] Chang, S., Tang, G.: A basis for augmentation quotients of finite Abelian groups. J. Algebra 327 (2011), 466-488. | DOI | MR | Zbl

[6] Magurn, B. A.: An Algebraic Introduction to $K$-Theory. Encyclopedia of Mathematics and Its Applications 87 Cambridge University Press, Cambridge (2002). | MR | Zbl

[7] Parmenter, M. M.: A basis for powers of the augmentation ideal. Algebra Colloq. 8 (2001), 121-128. | MR | Zbl

[8] Tang, G.: Presenting powers of augmentation ideals of elementary $p$-groups. $K$-Theory 23 (2001), 31-39. | DOI | MR | Zbl

[9] Tang, G.: On a problem of Karpilovsky. Algebra Colloq. 10 (2003), 11-16. | DOI | MR | Zbl

[10] Tang, G.: Structure of augmentation quotients of finite homocyclic Abelian groups. Sci. China, Ser. A. 50 (2007), 1280-1288. | DOI | MR | Zbl

[11] Wu, H., Tang, G. P.: Structure of powers of the augmentation ideal and their consecutive quotients for the Burnside ring of a finite abelian group. Adv. Math. (China) 36 (2007), 627-630 Chinese. English summary. | MR

Cité par Sources :