Results 1 to 5 of 5

Math Help - intersection of finitely generated groups

  1. #1
    Member
    Joined
    May 2008
    Posts
    75

    intersection of finitely generated groups

    Hi! We have a free group F and two finitely generated subgroups G and H. I want to show that their intersection is also finitely generated. The hint is to consider two graphs whose fundamental groups are G and H respectively. Then with these two graphs one can construct a graph whose fundamental group is exactly the intersection of G and H.

    Can anybody help me with this? I am completely stuck.

    nice greetings
    banach
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    What do you know already? Do you know about immersions (maps from a graph to another graph where the induced maps between edges starting at a given vertices is injective)?

    Basically, as your subgroups are finitely generated they correspond to immersions. Pullbacks of immersions correspond to intersections, and so construct the pullback of the two given immersions (and draw the diagram!). This pullback is a finite graph, and a component of it corresponds to your intersection. This gives you an immersion, and so a finite basis of your intersection has been found.

    I would say more, but I don't know what you already know...however, I think what I've said should be hint enough - it is quite a deep theorem so really you should know what a pullback etc. is if you're to tackle it...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    May 2008
    Posts
    75
    Thanks for your answer but i unfortunately i am not familiar with these terms. However, i think i know how to work it out now:
    Wlog we may assume that F=F(a,b). Let \Gamma_1 and \Gamma_2 the associated graphs of G and H. Define the graph \Gamma by V(\Gamma):=V(\Gamma_1) \times V(\Gamma_2) and there is an edge between (a,x) and (b,y) iff there are edges from a to b in \Gamma_1 and x to y in \Gamma_2 having the same label. Since G and H are f.g. the graphs \Gamma_1 and \Gamma_2 are finite graphs, thus \Gamma is also a finite graph whose fundamental group is G \cap H. Hence the intersection is f.g., qed.
    What do you think of that proof?

    Greetings
    Banach
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    Quote Originally Posted by Banach View Post
    Thanks for your answer but i unfortunately i am not familiar with these terms. However, i think i know how to work it out now:
    Wlog we may assume that F=F(a,b). Let \Gamma_1 and \Gamma_2 the associated graphs of G and H. Define the graph \Gamma by V(\Gamma):=V(\Gamma_1) \times V(\Gamma_2) and there is an edge between (a,x) and (b,y) iff there are edges from a to b in \Gamma_1 and x to y in \Gamma_2 having the same label. Since G and H are f.g. the graphs \Gamma_1 and \Gamma_2 are finite graphs, thus \Gamma is also a finite graph whose fundamental group is G \cap H. Hence the intersection is f.g., qed.
    What do you think of that proof?

    Greetings
    Banach
    That seems right, and is a much neater way than the way I was thinking of...
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    Quote Originally Posted by Banach View Post
    Thanks for your answer but i unfortunately i am not familiar with these terms. However, i think i know how to work it out now:
    Wlog we may assume that F=F(a,b). Let \Gamma_1 and \Gamma_2 the associated graphs of G and H. Define the graph \Gamma by V(\Gamma):=V(\Gamma_1) \times V(\Gamma_2) and there is an edge between (a,x) and (b,y) iff there are edges from a to b in \Gamma_1 and x to y in \Gamma_2 having the same label. Since G and H are f.g. the graphs \Gamma_1 and \Gamma_2 are finite graphs, thus \Gamma is also a finite graph whose fundamental group is G \cap H. Hence the intersection is f.g., qed.
    What do you think of that proof?

    Greetings
    Banach
    Actually, you would need to prove that the fundamental group of this graph is G \cap H. I do not think that this is obvious.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finitely Generated Abelian Groups
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: March 4th 2012, 05:32 AM
  2. finitely generated groups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: July 17th 2009, 04:09 PM
  3. free groups, finitely generated groups
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: May 23rd 2009, 03:31 AM
  4. finitely generated Abelian groups
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: April 19th 2009, 02:34 AM
  5. Replies: 5
    Last Post: January 22nd 2007, 07:51 PM

Search Tags


/mathhelpforum @mathhelpforum