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







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  • 5




    Hint: $A$ is a subgroup.
    – Wojowu
    2 days ago










  • Remember that $phi(g^{-1}) = (phi(g))^{-1}$.
    – Lucas Corrêa
    2 days ago










  • Thank you @Wojowu! Pretty trivial indeed with that hint.
    – Adar Gutman
    2 days ago

















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







share|cite|improve this question




















  • 5




    Hint: $A$ is a subgroup.
    – Wojowu
    2 days ago










  • Remember that $phi(g^{-1}) = (phi(g))^{-1}$.
    – Lucas Corrêa
    2 days ago










  • Thank you @Wojowu! Pretty trivial indeed with that hint.
    – Adar Gutman
    2 days ago















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







share|cite|improve this question















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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 days ago









Micah

29.4k1363104




29.4k1363104










asked 2 days ago









Adar Gutman

726




726








  • 5




    Hint: $A$ is a subgroup.
    – Wojowu
    2 days ago










  • Remember that $phi(g^{-1}) = (phi(g))^{-1}$.
    – Lucas Corrêa
    2 days ago










  • Thank you @Wojowu! Pretty trivial indeed with that hint.
    – Adar Gutman
    2 days ago
















  • 5




    Hint: $A$ is a subgroup.
    – Wojowu
    2 days ago










  • Remember that $phi(g^{-1}) = (phi(g))^{-1}$.
    – Lucas Corrêa
    2 days ago










  • Thank you @Wojowu! Pretty trivial indeed with that hint.
    – Adar Gutman
    2 days ago










5




5




Hint: $A$ is a subgroup.
– Wojowu
2 days ago




Hint: $A$ is a subgroup.
– Wojowu
2 days ago












Remember that $phi(g^{-1}) = (phi(g))^{-1}$.
– Lucas Corrêa
2 days ago




Remember that $phi(g^{-1}) = (phi(g))^{-1}$.
– Lucas Corrêa
2 days ago












Thank you @Wojowu! Pretty trivial indeed with that hint.
– Adar Gutman
2 days ago






Thank you @Wojowu! Pretty trivial indeed with that hint.
– Adar Gutman
2 days ago

















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