Given a finite group G, its double Burnside ring B(G,G), has a natural duality operation that arises from considering opposite (G,G)-bisets. In this article, we systematically study the subgroup of units of B(G,G), where elements are inverse to their dual, so called orthogonal units. We show the existence of an inflation map that embeds the group of orthogonal units of B(G/N,G/N) into the group of orthogonal units of B(G,G), when N is a normal subgroup of G, and study some properties and consequences. In particular, we use these maps to determine the orthogonal units of B(G,G), when G is a cyclic p-group, and p is an odd prime.

中文翻译：

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双Burnside环的正交单位

给定一个有限的群G，它的双Burnside环B（G，G）具有自然的对偶运算，这是通过考虑相对的（G，G）对偶而产生的。在本文中，我们系统地研究了B（G，G）单元的子组，其中元素与其对偶的逆，即正交单元。我们示出了充气地图嵌入的组的正交单位的存在乙（ģ / Ñ，ģ / Ñ）成团的正交单元的乙（G，G），当N是G的一个正常子集时，研究一些性质和后果。特别地，当G为循环p-群且p为奇质数时，我们使用这些映射图确定B（G，G）的正交单元。