I Why must the group N be finite in this result?

1,456
44
Ffom exercise 27 of Dummite and Foote: Let ##N## be a finite subgroup of ##G##. Show that ##gNg^{-1}\subseteq N## if and only if ##gNg^{-1} = N##.

Why must the subgroup ##N## be finite? Isn't this result true for subgroups of any size?
 

fresh_42

Mentor
Insights Author
2018 Award
11,561
8,015
Ffom exercise 27 of Dummite and Foote: Let ##N## be a finite subgroup of ##G##. Show that ##gNg^{-1}\subseteq N## if and only if ##gNg^{-1} = N##.

Why must the subgroup ##N## be finite? Isn't this result true for subgroups of any size?
Sure. It is even true for any subset ##N## ... as long as the quantor at ##g## is an ##\forall##.
 
1,456
44
Sure. It is even true for any subset ##N## ... as long as the quantor at ##g## is an ##\forall##.
Then why does the exercise bother with specifying that ##N## should be finite?
 

fresh_42

Mentor
Insights Author
2018 Award
11,561
8,015
Then why does the exercise bother with specifying that ##N## should be finite?
I have no idea. Given ##gSg^{-1}\subseteq S## for all ##g\in G## and a subset ##S\subseteq G##, we have especially for ##g^{-1}\, : \,g^{-1}S(g^{-1})^{-1} = g^{-1}Sg \subseteq S## and thus ##S \subseteq gSg^{-1} \subseteq S##.

Maybe the rest of the exercise depends on it. Here we only use the all quantor on the elements of ##G## and that they have an inverse. We don't even need ##S\subseteq G## as long as ##G## operates via conjugation on a set ##S##.
 

fresh_42

Mentor
Insights Author
2018 Award
11,561
8,015

martinbn

Science Advisor
1,543
399
##gNg^{-1}## and ##N## have the same index.
 

Want to reply to this thread?

"Why must the group N be finite in this result?" You must log in or register to reply here.

Related Threads for: Why must the group N be finite in this result?

  • Posted
Replies
1
Views
2K
  • Posted
Replies
2
Views
3K
  • Posted
Replies
3
Views
2K
  • Posted
Replies
2
Views
1K
  • Posted
Replies
1
Views
1K
Replies
16
Views
2K
Replies
5
Views
2K
Replies
4
Views
2K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top