Automorphism acting trivially on more than half the elements of the group is trivial
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Let $G$ be a group of finite order, wih $phi:Grightarrow G$ an automorphism on $G$.
Assume $A=$ {$ gin G|phi(g)=g $} consists of more than half of the elements of $G$.
In other words, $|A| > |G|/2$.
Prove $phi$ is the identity automorphism.
abstract-algebra group-theory
edited Nov 19 at 19:43
Micah
29.5k1363104
asked Nov 19 at 19:38
Adar Gutman
726
add a comment |
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1
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Let $G$ be a group of finite order, wih $phi:Grightarrow G$ an automorphism on $G$.
Assume $A=$ {$ gin G|phi(g)=g $} consists of more than half of the elements of $G$.
In other words, $|A| > |G|/2$.
Prove $phi$ is the identity automorphism.
abstract-algebra group-theory
edited Nov 19 at 19:43
Micah
29.5k1363104
asked Nov 19 at 19:38
Adar Gutman
726
5
Hint: $A$ is a subgroup.
– Wojowu
Nov 19 at 19:42
Remember that $phi(g^{-1}) = (phi(g))^{-1}$.
– Lucas Corrêa
Nov 19 at 19:45
Thank you @Wojowu! Pretty trivial indeed with that hint.
– Adar Gutman
Nov 19 at 19:46
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $G$ be a group of finite order, wih $phi:Grightarrow G$ an automorphism on $G$.
Assume $A=$ {$ gin G|phi(g)=g $} consists of more than half of the elements of $G$.
In other words, $|A| > |G|/2$.
Prove $phi$ is the identity automorphism.
abstract-algebra group-theory
edited Nov 19 at 19:43
Micah
29.5k1363104
asked Nov 19 at 19:38
Adar Gutman
726
Let $G$ be a group of finite order, wih $phi:Grightarrow G$ an automorphism on $G$.
Assume $A=$ {$ gin G|phi(g)=g $} consists of more than half of the elements of $G$.
In other words, $|A| > |G|/2$.
Prove $phi$ is the identity automorphism.
abstract-algebra group-theory
abstract-algebra group-theory
edited Nov 19 at 19:43
Micah
29.5k1363104
asked Nov 19 at 19:38
Adar Gutman
726
edited Nov 19 at 19:43
Micah
29.5k1363104
asked Nov 19 at 19:38
Adar Gutman
726
edited Nov 19 at 19:43
Micah
29.5k1363104
edited Nov 19 at 19:43
Micah
29.5k1363104
edited Nov 19 at 19:43
Micah
29.5k1363104
29.5k1363104
asked Nov 19 at 19:38
Adar Gutman
726
asked Nov 19 at 19:38
Adar Gutman
726
asked Nov 19 at 19:38
Adar Gutman
726
726
5
Hint: $A$ is a subgroup.
– Wojowu
Nov 19 at 19:42
Remember that $phi(g^{-1}) = (phi(g))^{-1}$.
– Lucas Corrêa
Nov 19 at 19:45
Thank you @Wojowu! Pretty trivial indeed with that hint.
– Adar Gutman
Nov 19 at 19:46
add a comment |
5
Hint: $A$ is a subgroup.
– Wojowu
Nov 19 at 19:42
Remember that $phi(g^{-1}) = (phi(g))^{-1}$.
– Lucas Corrêa
Nov 19 at 19:45
Thank you @Wojowu! Pretty trivial indeed with that hint.
– Adar Gutman
Nov 19 at 19:46
5
5
Hint: $A$ is a subgroup.
– Wojowu
Nov 19 at 19:42
Hint: $A$ is a subgroup.
– Wojowu
Nov 19 at 19:42
Remember that $phi(g^{-1}) = (phi(g))^{-1}$.
– Lucas Corrêa
Nov 19 at 19:45
Remember that $phi(g^{-1}) = (phi(g))^{-1}$.
– Lucas Corrêa
Nov 19 at 19:45
Thank you @Wojowu! Pretty trivial indeed with that hint.
– Adar Gutman
Nov 19 at 19:46
Thank you @Wojowu! Pretty trivial indeed with that hint.
– Adar Gutman
Nov 19 at 19:46
add a comment |
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5
Hint: $A$ is a subgroup.
– Wojowu
Nov 19 at 19:42
Remember that $phi(g^{-1}) = (phi(g))^{-1}$.
– Lucas Corrêa
Nov 19 at 19:45
Thank you @Wojowu! Pretty trivial indeed with that hint.
– Adar Gutman
Nov 19 at 19:46