Mixing
inner automorphisms unequal to the inverse except at identity
$begingroup$
Let $N$ be a normal subgroup of a finite group $G$. Then for each $g in G$ there exists an automorphism of $N$ given by $phi_g(n)=g^{-1}ng$. Suppose we impose the following condition: $$phi_g(n) neq n^{-1},~~~forall gin G,~~~ forall n in N backslash {1}.$$
What I can show so far is that for this condition to hold, it is necessary that $N$ have odd order. If $N$ is contained in the center of $G$, this is also sufficient.
My question is that whether you have seen such a condition before in group theory literature, and if not, what would be an appropriate terminology to use for such a normal subgroup?
group-theory terminology
asked Jan 31 at 20:22
MarcoMarco
1,981112
$endgroup$
add a comment |
$begingroup$
Let $N$ be a normal subgroup of a finite group $G$. Then for each $g in G$ there exists an automorphism of $N$ given by $phi_g(n)=g^{-1}ng$. Suppose we impose the following condition: $$phi_g(n) neq n^{-1},~~~forall gin G,~~~ forall n in N backslash {1}.$$
What I can show so far is that for this condition to hold, it is necessary that $N$ have odd order. If $N$ is contained in the center of $G$, this is also sufficient.
My question is that whether you have seen such a condition before in group theory literature, and if not, what would be an appropriate terminology to use for such a normal subgroup?
group-theory terminology
asked Jan 31 at 20:22
MarcoMarco
1,981112
$endgroup$
2
$begingroup$
I've seen some works on checking groups in which no element is automorphic to its inverse ("automorphic to its inverse" means there exists $phiinmathrm{Aut}(G)$ such that $phi(n)=n^{-1}$). Since $G/C_G(N)$ embeds into $mathrm{Aut}(N)$, this looks related.
$endgroup$
– Arturo Magidin
Feb 1 at 17:54
$begingroup$
@ArturoMagidin that definitely seems related. Would you be able to give any reference where these groups are mentioned?
$endgroup$
– Marco
Feb 3 at 20:23
add a comment |
$begingroup$
Let $N$ be a normal subgroup of a finite group $G$. Then for each $g in G$ there exists an automorphism of $N$ given by $phi_g(n)=g^{-1}ng$. Suppose we impose the following condition: $$phi_g(n) neq n^{-1},~~~forall gin G,~~~ forall n in N backslash {1}.$$
What I can show so far is that for this condition to hold, it is necessary that $N$ have odd order. If $N$ is contained in the center of $G$, this is also sufficient.
My question is that whether you have seen such a condition before in group theory literature, and if not, what would be an appropriate terminology to use for such a normal subgroup?
group-theory terminology
asked Jan 31 at 20:22
MarcoMarco
1,981112
$endgroup$
Let $N$ be a normal subgroup of a finite group $G$. Then for each $g in G$ there exists an automorphism of $N$ given by $phi_g(n)=g^{-1}ng$. Suppose we impose the following condition: $$phi_g(n) neq n^{-1},~~~forall gin G,~~~ forall n in N backslash {1}.$$
What I can show so far is that for this condition to hold, it is necessary that $N$ have odd order. If $N$ is contained in the center of $G$, this is also sufficient.
My question is that whether you have seen such a condition before in group theory literature, and if not, what would be an appropriate terminology to use for such a normal subgroup?
group-theory terminology
group-theory terminology
asked Jan 31 at 20:22
MarcoMarco
1,981112
asked Jan 31 at 20:22
MarcoMarco
1,981112
asked Jan 31 at 20:22
MarcoMarco
1,981112
asked Jan 31 at 20:22
MarcoMarco
1,981112
asked Jan 31 at 20:22
MarcoMarco
1,981112
1,981112
2
$begingroup$
I've seen some works on checking groups in which no element is automorphic to its inverse ("automorphic to its inverse" means there exists $phiinmathrm{Aut}(G)$ such that $phi(n)=n^{-1}$). Since $G/C_G(N)$ embeds into $mathrm{Aut}(N)$, this looks related.
$endgroup$
– Arturo Magidin
Feb 1 at 17:54
$begingroup$
@ArturoMagidin that definitely seems related. Would you be able to give any reference where these groups are mentioned?
$endgroup$
– Marco
Feb 3 at 20:23
add a comment |
2
$begingroup$
I've seen some works on checking groups in which no element is automorphic to its inverse ("automorphic to its inverse" means there exists $phiinmathrm{Aut}(G)$ such that $phi(n)=n^{-1}$). Since $G/C_G(N)$ embeds into $mathrm{Aut}(N)$, this looks related.
$endgroup$
– Arturo Magidin
Feb 1 at 17:54
$begingroup$
@ArturoMagidin that definitely seems related. Would you be able to give any reference where these groups are mentioned?
$endgroup$
– Marco
Feb 3 at 20:23
2
2
$begingroup$
I've seen some works on checking groups in which no element is automorphic to its inverse ("automorphic to its inverse" means there exists $phiinmathrm{Aut}(G)$ such that $phi(n)=n^{-1}$). Since $G/C_G(N)$ embeds into $mathrm{Aut}(N)$, this looks related.
$endgroup$
– Arturo Magidin
Feb 1 at 17:54
$begingroup$
I've seen some works on checking groups in which no element is automorphic to its inverse ("automorphic to its inverse" means there exists $phiinmathrm{Aut}(G)$ such that $phi(n)=n^{-1}$). Since $G/C_G(N)$ embeds into $mathrm{Aut}(N)$, this looks related.
$endgroup$
– Arturo Magidin
Feb 1 at 17:54
$begingroup$
@ArturoMagidin that definitely seems related. Would you be able to give any reference where these groups are mentioned?
$endgroup$
– Marco
Feb 3 at 20:23
$begingroup$
@ArturoMagidin that definitely seems related. Would you be able to give any reference where these groups are mentioned?
$endgroup$
– Marco
Feb 3 at 20:23
add a comment |
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2
$begingroup$
I've seen some works on checking groups in which no element is automorphic to its inverse ("automorphic to its inverse" means there exists $phiinmathrm{Aut}(G)$ such that $phi(n)=n^{-1}$). Since $G/C_G(N)$ embeds into $mathrm{Aut}(N)$, this looks related.
$endgroup$
– Arturo Magidin
Feb 1 at 17:54
$begingroup$
@ArturoMagidin that definitely seems related. Would you be able to give any reference where these groups are mentioned?
$endgroup$
– Marco
Feb 3 at 20:23