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Augmentation quotients for Burnside rings of generalized dihedral groups

Shan Chang — 2016

Czechoslovak Mathematical Journal

Let H be a finite abelian group of odd order, 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 Ω ( 𝒟 ) be the Burnside ring of 𝒟 , Δ ( 𝒟 ) be the augmentation ideal of Ω ( 𝒟 ) . Denote by Δ n ( 𝒟 ) and Q n ( 𝒟 ) the n th power of Δ ( 𝒟 ) and the n th consecutive quotient group Δ n ( 𝒟 ) / Δ n + 1 ( 𝒟 ) , respectively. This paper provides an explicit -basis for Δ n ( 𝒟 ) and determines the isomorphism class of Q n ( 𝒟 ) for each positive integer n .

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