A rigid automorphism of a linking system is an automorphism that restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $2$ is elementary abelian and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group G restricts to the identity on the centric linking system for G, then it is of $p'$ -order modulo the group of inner automorphisms, provided G has no nontrivial normal $p'$ -subgroups. We present two applications of this last result, one to tame fusion systems.

中文翻译：

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链接系统的刚性自同构

链接系统的刚性自同构是限制在 Sylow 子群上的恒等式的自同构。刚性内自同构是 Sylow 子群中心元素的共轭。在奇数素数处，已知中心链接系统的每个刚性自同构都是内部的。我们证明了素数连接系统的刚性外自同构群 $2$ 是初等阿贝尔，并且它在刚性内自同构子群上分裂。在第二个结果中，我们证明如果一个有限群的自同构G限制在中心链接系统上的身份G，那么它是 $p'$ -order 模内部自同构群，提供G没有非平凡的正常 $p'$ -亚组。我们提出了最后一个结果的两种应用，一种是驯服融合系统。