An induction theorem for the unit groups of Burnside rings of 2-groups
An induction theorem for the unit groups of Burnside rings of 2-groups
复制标题
2-群Burnside环单位群的归纳定理
DOI：
10.1016/j.jalgebra.2005.03.029
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发表时间：
2005
影响因子：
0.9
通讯作者：
Ergün Yalçın
中科院分区：
文献类型：
--
作者：
Ergün Yalçın
设G为2-群，B(G)×denote为G的Burnside环的单位群。对于G的每一个子商H/K，有一个来自B（H/K）×to B(G)×defined的广义归纳图作为膨胀和乘法归纳图的组合。我们证明了广义归纳的乘积映射∏B(H/K) x→B(G)×is满射，当乘积被接管与平凡群同构的所有子商的集合或与n大于或等于4的2n阶的二面体2-群。作为应用，我们给出了Tornehave [2-group的Burnside环的单位群，Aarhus Universitet Preprint series 1983/84 41,1984年5月]定理的代数证明，该定理表明tom Dieck的指数映射从G到B(G)×is满射。给出了G的Burnside环到B(G) x的指数映射满射的一个充分条件。
Let G be a 2-group and B(G)×denote the group of units of the Burnside ring of G. For each subquotient H/K of G, there is a generalized induction map from B(H/K)×to B(G)×defined as the composition of inflation and multiplicative induction maps. We prove that the product of generalized induction maps ∏B(H/K)×→B(G)×is surjective when the product is taken over the set of all subquotients that are isomorphic to the trivial group or a dihedral 2-group of order 2nwith n⩾4. As an application, we give an algebraic proof for a theorem by Tornehave [The unit group for the Burnside ring of a 2-group, Aarhus Universitet Preprint series 1983/84 41, May 1984] which states that tom Dieck's exponential map from the real representation ring of G to B(G)×is surjective. We also give a sufficient condition for the surjectivity of the exponential map from the Burnside ring of G to B(G)×.