Abstract. We show that the 20 graph Heawood family, obtained by a combination of ∇Y and Y∇ moves on K 7 , is precisely the set of graphs of at most 21 edges that are minor minimal for the property not 2-apex. As a corollary, this gives a new proof that the 14 graphs obtained by ∇Y moves on K 7 are the minor minimal intrinsically knotted graphs of 21 or fewer edges. Similarly, we argue that the seven graph Petersen family, obtained from K 6 , is the set of graphs of at most 17 edges that are minor minimal for the property not apex.
We show that the 14 graphs obtained by ∇Y moves on K 7 constitute a complete list of the minor minimal intrinsically knotted graphs on 21 edges. We also present evidence in support of a conjecture that the 20 graph Heawood family, obtained by a combination of ∇Y and Y∇ moves on K 7 , is the list of graphs of size 21 that are minor minimal with respect to the property not 2-apex.
The theory of bisets has been very useful in progress towards settling the longstanding question of determining units for the Burnside ring. In 2006 Bouc used bisets to settle the question for p-groups. In this paper, we provide a standard basis for the unit group of the Burnside ring for groups that contain a abelian subgroups of index two. We then extend this result to groups G, where G has a normal subgroup, N , of odd index, such that N contains an abelian subgroups of index 2. Next, we study the structure of the unit group of the Burnside ring as a biset functor, B × on this class of groups and determine its lattice of subfunctors. We then use this to determine the composition factors of B × over this class of groups. Additionally, we give a sufficient condition for when the functor B × , defined on a class of groups closed under subquotients, has uncountably many subfunctors.
AbstractFor a saturated fusion system {\mathcal{F}} on a p-group S, we study the Burnside ring of the fusion system {B(\mathcal{F})}, as defined by Matthew Gelvin and Sune Reeh, which is a subring of the Burnside ring {B(S)}.
We give criteria for an element of {B(S)} to be in {B(\mathcal{F})} determined by the {\mathcal{F}}-automorphism groups of essential subgroups of S.
When {\mathcal{F}} is the fusion system induced by a finite group G with S as a Sylow p-group, we show that the restriction of {B(G)} to {B(S)} has image equal to {B(\mathcal{F})}.
We also show that, for {p=2}, we can gain information about the fusion system by studying the unit group {B(\mathcal{F})^{\times}}.
When S is abelian, we completely determine this unit group.
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