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Math Help - subgroups

  1. #1
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    subgroups

    what does it mean by to be a subgroup H of G where G is a group, then for all a,b elements of H, a^-1 b is an element of H.

    i thought that to be a subgroup, elements of H have to be in G and it has to obey the conditions of a group?

    thanks
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by alexandrabel90 View Post
    what does it mean by to be a subgroup H of G where G is a group, then for all a,b elements of H, a^-1 b is an element of H.

    i thought that to be a subgroup, elements of H have to be in G and it has to obey the conditions of a group?
    You are right, and that would be the most natural way of defining H to be a subgroup of G. But requiring for all a,b\in H that a^{-1}b\in H is just an equivalent, and apparently some think more useful way of defining the same concept (more useful in the sense that proving a subset H of G to be a subgroup requires proving only that single statement).

    Assuming that funny definition of being a subgroup, if you set b=a you get that e=a^{-1}a\in H; thus H contains the neutral element e. And once you have shown that you can set b=e and get that H also contains the inverse element of every one of its elements: a^{-1}=a^{-1}e\in H.
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