Burnside rings of fusion systems and their unit groups

Abstract

<jats:title>Abstract</jats:title><jats:p>For a saturated fusion system <jats:inline-formula id="j_jgth-2019-0145_ineq_9999_w2aab3b7e1973b1b6b1aab1c15b1b1Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0145_eq_0380.png" /><jats:tex-math>{\mathcal{F}}</jats:tex-math></jats:alternatives></jats:inline-formula> on a <jats:italic>p</jats:italic>-group <jats:italic>S</jats:italic>, we study the Burnside ring of the fusion system <jats:inline-formula id="j_jgth-2019-0145_ineq_9998_w2aab3b7e1973b1b6b1aab1c15b1b7Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0145_eq_0166.png" /><jats:tex-math>{B(\mathcal{F})}</jats:tex-math></jats:alternatives></jats:inline-formula>, as defined by Matthew Gelvin and Sune Reeh, which is a subring of the Burnside ring <jats:inline-formula id="j_jgth-2019-0145_ineq_9997_w2aab3b7e1973b1b6b1aab1c15b1b9Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0145_eq_0145.png" /><jats:tex-math>{B(S)}</jats:tex-math></jats:alternatives></jats:inline-formula>. We give criteria for an element of <jats:inline-formula id="j_jgth-2019-0145_ineq_9996_w2aab3b7e1973b1b6b1aab1c15b1c11Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0145_eq_0145.png" /><jats:tex-math>{B(S)}</jats:tex-math></jats:alternatives></jats:inline-formula> to be in <jats:inline-formula id="j_jgth-2019-0145_ineq_9995_w2aab3b7e1973b1b6b1aab1c15b1c13Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0145_eq_0166.png" /><jats:tex-math>{B(\mathcal{F})}</jats:tex-math></jats:alternatives></jats:inline-formula> determined by the <jats:inline-formula id="j_jgth-2019-0145_ineq_9994_w2aab3b7e1973b1b6b1aab1c15b1c15Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0145_eq_0380.png" /><jats:tex-math>{\mathcal{F}}</jats:tex-math></jats:alternatives></jats:inline-formula>-automorphism groups of essential subgroups of <jats:italic>S</jats:italic>. When <jats:inline-formula id="j_jgth-2019-0145_ineq_9993_w2aab3b7e1973b1b6b1aab1c15b1c19Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0145_eq_0380.png" /><jats:tex-math>{\mathcal{F}}</jats:tex-math></jats:alternatives></jats:inline-formula> is the fusion system induced by a finite group <jats:italic>G</jats:italic> with <jats:italic>S</jats:italic> as a Sylow <jats:italic>p</jats:italic>-group, we show that the restriction of <jats:inline-formula id="j_jgth-2019-0145_ineq_9992_w2aab3b7e1973b1b6b1aab1c15b1c27Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0145_eq_0136.png" /><jats:tex-math>{B(G)}</jats:tex-math></jats:alternatives></jats:inline-formula> to <jats:inline-formula id="j_jgth-2019-0145_ineq_9991_w2aab3b7e1973b1b6b1aab1c15b1c29Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0145_eq_0145.png" /><jats:tex-math>{B(S)}</jats:tex-math></jats:alternatives></jats:inline-formula> has image equal to <jats:inline-formula id="j_jgth-2019-0145_ineq_9990_w2aab3b7e1973b1b6b1aab1c15b1c31Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0145_eq_0166.png" /><jats:tex-math>{B(\mathcal{F})}</jats:tex-math></jats:alternatives></jats:inline-formula>. We also show that, for <jats:inline-formula id="j_jgth-2019-0145_ineq_9989_w2aab3b7e1973b1b6b1aab1c15b1c33Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0145_eq_0502.png" /><jats:tex-math>{p=2}</jats:tex-math></jats:alternatives></jats:inline-formula>, we can gain information about the fusion system by studying the unit group <jats:inline-formula id="j_jgth-2019-0145_ineq_9988_w2aab3b7e1973b1b6b1aab1c15b1c35Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0145_eq_0165.png" /><jats:tex-math>{B(\mathcal{F})^{\times}}</jats:tex-math></jats:alternatives></jats:inline-formula>. When <jats:italic>S</jats:italic> is abelian, we completely determine this unit group.</jats:p>