Proof Using Def. of Groups and Def. of Subgroups

  1. Hello,

    This is my question:

    (i) Let H and K be subgroups of a group G. Prove that the intersection of H and K is also a subgroup of G.

    (ii) Give an example, using suitable sugroups of the goup of integers with the operation addition, (Z,+), to show that if H and K are subgroups of a group G, then the union of U and K need not be a subgroup of G.


    I figure that if I can do (i) then (ii) will follow. But I am unsure of how to do (i). (In previous assignments I did proofs on subsets, unions, intersections of sets but I did poorly on them.)

    I posted a previous question stating the four properties of a set of which all must be satisfied to be defined as group. From my understanding of subgroups, only two of these properties must be satisfied to define a subgroup of a group.

    Definition: A subgroup of a group G (G,o) is any nonempty subset H of G such that H is a group with the same operation, o. To check that H is a subgroup, verify the following:

    S1: If x and y are elements of H, then x o y is an element of H.
    s2: If x is an element of H, the the inverse is an element of H.


    If H and K are subgroups then they must satisfy the aforementioned properties. And as a consequence if S1 and S2 are satisfied, and because H and K are elements of G, the remaining 2 properties that define a group are satisfied as well.

    I also know that by definition, H intersects K iff some of the elements of K are also elements of H.

    But I am unsure of where to go from here, much less construct a proof that proves that the intersection of H and K is also a subgroup of G.

    (After typing this I think that I have some idea so I might be back later to submit more ideas/work regarding this question).

    Any help to steer me in the right direction would be appreciated. Thankyou.
     
  2. jcsd
  3. Tom Mattson

    Tom Mattson 5,539
    Staff Emeritus
    Science Advisor
    Gold Member

  4. Thanks Tom. I might get back to you if I don't understand a step or few. Or for some clarification of some concept. That happens from time to time you know.

    Thanks again.
     
  5. HallsofIvy

    HallsofIvy 40,382
    Staff Emeritus
    Science Advisor

    But it's not just a matter of WHETHER they intersect but WHAT the intersection is.


    Yes, exactly. Suppose x and y are members of H intersect K.
    Then x and y are members of ____ and, since ____ is a subgroup, x+y is in ____.

    Since x and y are members of H intersect K, they also are members of ___ which is a subgroup. Therefore, x+y is a member of _____.

    Since x+y is in ___ and ___, it is in ___ intersect ___.

    Fill in the blanks!:smile:
     
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