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Math Help - Union of Subgroups

  1. #1
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    Union of Subgroups

    True or False:

    If H and K are subgroups of G, then H \cap K is a subgroup of G.
    If H_{t} is a subgroup of G for all t \in I, then \cap_{t \in I} H_{t} is a subgroup of G.

    Explanations would be greatly appreciated.
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  2. #2
    Senior Member Dinkydoe's Avatar
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    You only need to verify that these sets are groups,

    that is
    (1)  e\in K\cap H
    (2) a\in K\cap H \Rightarrow a^{-1}\in K \cap H
    ((3) associativity is automatically satisfied)

    These are trivial exercises. Keep the definition of intersections in mind.
    The same goes for \bigcap_{t\in I} H_t

    (Edit: What you wrote was not the right symbol for Union. The union symbol is \bigcup. However the excercise is the same)
    Last edited by Dinkydoe; March 17th 2010 at 02:20 PM.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Dinkydoe View Post

    (Edit: What you wrote was not the right symbol for Union. The union symbol is \bigcup. However the excercise is the same)
    I disagree. The union of two subgroups need not be a subgroup. For example

    2\mathbb{Z}\cup3\mathbb{Z}\not\leqslant\mathbb{Z}
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  4. #4
    Senior Member Dinkydoe's Avatar
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    Yes, true. However on second thought, the poster mustve meant -intersection of sub-groups-
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  5. #5
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    My mistake, I did mean to say intersection of Subgroups
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