On $n$th class-preserving automorphism of finite $p$-group
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Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $alpha$ of $G$ is called an $n$th class-preserving if for each $xin G$, there exists an element $g_xin gamma_n(G)$ such that $alpha(x)=g_x^{-1}xg_x$, where $gamma_n(G)$ denotes the $n$th term of the lower central series of $G$. An automorphism $alpha$ of $G$ is called a central automorphism if $x^{-1}alpha(x)in Z(G)$ for all $xin G$. Let $Aut_{c}^n(G)$ and $Autcent(G)$ respectively denote the group of all $n$th class-preserving and central automorphisms of $G$.
My question is the following: Give some examples of finite non-abelian $p$-group $G$ of nilpotency class 3 such that $Aut_{c}^2(G)=Autcent(G)$.
abstract-algebra group-theory finite-groups p-groups
asked Dec 8 '18 at 4:58
RohitRohit
213
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$begingroup$
Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $alpha$ of $G$ is called an $n$th class-preserving if for each $xin G$, there exists an element $g_xin gamma_n(G)$ such that $alpha(x)=g_x^{-1}xg_x$, where $gamma_n(G)$ denotes the $n$th term of the lower central series of $G$. An automorphism $alpha$ of $G$ is called a central automorphism if $x^{-1}alpha(x)in Z(G)$ for all $xin G$. Let $Aut_{c}^n(G)$ and $Autcent(G)$ respectively denote the group of all $n$th class-preserving and central automorphisms of $G$.
My question is the following: Give some examples of finite non-abelian $p$-group $G$ of nilpotency class 3 such that $Aut_{c}^2(G)=Autcent(G)$.
abstract-algebra group-theory finite-groups p-groups
asked Dec 8 '18 at 4:58
RohitRohit
213
$endgroup$
add a comment |
$begingroup$
Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $alpha$ of $G$ is called an $n$th class-preserving if for each $xin G$, there exists an element $g_xin gamma_n(G)$ such that $alpha(x)=g_x^{-1}xg_x$, where $gamma_n(G)$ denotes the $n$th term of the lower central series of $G$. An automorphism $alpha$ of $G$ is called a central automorphism if $x^{-1}alpha(x)in Z(G)$ for all $xin G$. Let $Aut_{c}^n(G)$ and $Autcent(G)$ respectively denote the group of all $n$th class-preserving and central automorphisms of $G$.
My question is the following: Give some examples of finite non-abelian $p$-group $G$ of nilpotency class 3 such that $Aut_{c}^2(G)=Autcent(G)$.
abstract-algebra group-theory finite-groups p-groups
asked Dec 8 '18 at 4:58
RohitRohit
213
$endgroup$
Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $alpha$ of $G$ is called an $n$th class-preserving if for each $xin G$, there exists an element $g_xin gamma_n(G)$ such that $alpha(x)=g_x^{-1}xg_x$, where $gamma_n(G)$ denotes the $n$th term of the lower central series of $G$. An automorphism $alpha$ of $G$ is called a central automorphism if $x^{-1}alpha(x)in Z(G)$ for all $xin G$. Let $Aut_{c}^n(G)$ and $Autcent(G)$ respectively denote the group of all $n$th class-preserving and central automorphisms of $G$.
My question is the following: Give some examples of finite non-abelian $p$-group $G$ of nilpotency class 3 such that $Aut_{c}^2(G)=Autcent(G)$.
abstract-algebra group-theory finite-groups p-groups
abstract-algebra group-theory finite-groups p-groups
asked Dec 8 '18 at 4:58
RohitRohit
213
asked Dec 8 '18 at 4:58
RohitRohit
213
asked Dec 8 '18 at 4:58
RohitRohit
213
asked Dec 8 '18 at 4:58
RohitRohit
213
asked Dec 8 '18 at 4:58
RohitRohit
213
213
add a comment |
add a comment |
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