2013年10月17日 · A nice set of generators for the automorphism group of a finite abelian group is described by Garrett Birkhoff in his paper titled "Subgroups of abelian groups", Proc. London Math. Soc., s2-38(1):385-401, 1935 .

2015年12月8日 · $\begingroup$ That's not true either -- such an automorphism always fixes a polarization on X, given by the restriction of the hyperplane from P^n. But general automorphisms don't fix any polarization; infinite order automorphisms of surfaces are a typical counterexample.

2015年8月29日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

2019年4月29日 · The object you've defined is not the group of automorphisms of $\mathbb{P}^n$; among other things, it is a group-valued functor, not a group.

2020年12月19日 · I am looking for an explicit example of a function field other than rational with an automorphism which fixes places of different degrees. Also, is there a counterexample, namely a function field with all the automorphisms don't fix any places of different degree.

How does this group object $\operatorname{Sym}(X)$, if it exists, relate to the automorphism group $\operatorname{Aut}_C(X)$ (which is an actual group, not a group object)? We can also define a "group object action by automorphisms" as the action of one group object on another satisfying the additional condition of being group automorphisms.

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The automorphism group $\mathrm{Aut}(\mathscr C)$ of a codebook is the subset of permutations on $\{1,2,\dotsc,n\}$ (which acts on $\Sigma^n$) that fix $\mathscr C$. Code designers prefer codes with more structures; so the alphabet $\Sigma = \mathbb F_q$ is usually taken to be finite fields, especially $\mathbb F_2$ ; less commonly the alphabet ...

2015年9月20日 · depending on whether the group is assumed to be the full automorphism group of the surface or only a subgroup of it. In my answer below I only assumed that the group was an automorphism group, not necessarily the full automorphism group. This may account for the discrepancy between my answer (there are two groups of order 32) and Dan Petersen's ...

2024年2月17日 · If $\sigma$ is an automorphism of order $6$, then $\sigma^{2}$ has $3$ fixed points (by conjugation on the simple group) and $\sigma^{3}$ has $6$ fixed points, while $\sigma$ has $9$ regular orbits. If $\tau$ is an automorphism of order $4$, then $\tau^{2}$ has $4$ fixed points, so $\tau$ has $14$ regular orbits.