Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Why is the core of a subgroup contained in the subgroup?

  1. Jan 23, 2015 #1
    Let H be a subgroup of G, then:
    Core H = {a in G | a is an element of gHg^(-1) for all g in G} = The intersection of all conjugates of H in G

    My book goes on to say that every element of Core H is in H itself because H is a conjugate to itself. Previously, I understood that H was a conjugate to itself because 1H1^(-1) = H, where 1 is the identity. However, the definition of Core H is the *intersection* of all conjugates, and it seems that core of H would be included in H only if gHg^(-1) = H for all g in G, (aka H is normal).

    What am I not understanding here? Anyone who can shed any light what so ever is greatly appreciated.
     
  2. jcsd
  3. Jan 23, 2015 #2

    lavinia

    User Avatar
    Science Advisor
    Gold Member

    The intersection of all of the conjugates must be in H since H is one of the conjugates. For instance, the identity element is in the intersection of all of the conjugates.
     
  4. Jan 23, 2015 #3
    I understand the identity element is in the intersection of all of the conjugates, but how would you prove that an arbitrary h in H is in the intersection?
     
  5. Jan 23, 2015 #4

    lavinia

    User Avatar
    Science Advisor
    Gold Member

    Not all of the elements of H are in the intersection - in general.

    For instance take the group with two generators a and b with the relation
    ## a^2 = b^4 = 1##
    ## aba = b^3##

    ##a## generates a subgroup,##H##, of order 2.

    ##H = {a,1}##

    ## bHb^{-1} = bab^{-1}, 1##

    but
    ## bab^{-1} = bab^{3} = ab^{2}##

    So a is not in the core of H
     
  6. Jan 23, 2015 #5
    Wow thanks, that really helps. Can you prove why anything in the core is also in H?
     
  7. Jan 23, 2015 #6

    lavinia

    User Avatar
    Science Advisor
    Gold Member

    I misread your question. Below I am explaining why not all of H need be in the core.

    Well the identity always is. But after that there are no guarantees. If H is a normal subgroup then all of H is in the core. In the example I gave - which BTW is the dihedral group of order 8 - I purposefully chose a not normal subgroup.

    Try to come up with and example where the core of H is greater that the identity element but not all of H.
     
    Last edited: Jan 23, 2015
  8. Jan 23, 2015 #7

    lavinia

    User Avatar
    Science Advisor
    Gold Member

    Sorry I misread your question. I was saying that not all of H needs to be in the core. The reason that the core must be contained in H is that His one of the sets in the intersection. So if isn't in H it is not in the core.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Why is the core of a subgroup contained in the subgroup?
  1. Order of subgroup/ (Replies: 1)

  2. Kernal of a subgroup (Replies: 8)

Loading...