Timeline for automorphisms of a finite $p$-group

5 events

when toggle format	what		by	license	comment
Jul 18, 2017 at 13:39	comment	added	YCor		OK, there are indeed 4 conventions for $[x,y]$, namely (writing $x'=x^{-1}$) $xyx'y'$, $yxy'x'$, $x'y'xy$, $y'x'yx$. For these conventions, the relation $[b,a]=b^p$ reads as resp. $aba'=b^{p+1}$, $aba'=b^{1-p}$, $a'ba=b^{1-p}$, $a'ba=b^{1+p}$. Since the order of both $1+p$ and $1-p$ modulo $p^{n+1}$ is exactly $p^n$ for odd $p$, all conventions yield the same group, namely the semidirect product $(\mathbf{Z}/p^{n+1}\mathbf{Z})\rtimes C$, where $C$ is the subgroup (of index $p-1$; cyclic of order $p^n$) in $(\mathbf{Z}/p^{n+1}\mathbf{Z})^\times$ consisting of those elements $\equiv 1$ mod $p$.
Jul 18, 2017 at 4:49	answer	added	Arturo Magidin		timeline score: 2
Jul 17, 2017 at 21:49	comment	added	Derek Holt		$a \mapsto a$, $b \mapsto b^{-1}$ defines an automorphism of order $2$, which must be non-inner. Computer calculations indicate that the automorphism group has order $(p-1)p^{2n+1}$, so the outer automorphism group has order $(p-1)p$. The subgroup of order $p-1$ is coming from automorphisms that fix $a$ and map $b$ to a power of itself.
Jul 17, 2017 at 20:54	comment	added	YCor		Which convention do you use for $[b,a]$?
Jul 17, 2017 at 20:37	history	asked	banoo	CC BY-SA 3.0