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Math Help - homomorphism

  1. #1
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    homomorphism

    a. Find all subgroups of S_3, and determine which are normal
    b. Find all subgroups of the quaternion group, and determine which are normal

    could anyone please help me?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by jin_nzzang View Post
    a. Find all subgroups of S_3, and determine which are normal
    b. Find all subgroups of the quaternion group, and determine which are normal

    could anyone please help me?
    S_3 has order 6 and so it can only have subgroup of order 1, 2, 3 or 6 (can you see why?).

    Now, <(123)> = <(132)> = A_3 is a subgroup of order 3. Just try every element of the group too see if this is normal or not - for example, (12)(123)(12)^{-1} = (12)(123)(12) = (132) \in A_3. Further, this is the only subgroup of order 3, as every element in S_3 that is not in A_3 has order 2, and so cannot be in a group of order 3 (the order of an element in a group MUST divide the order of the group.

    I will leave you too find the subgroups of order 2, and I will also leave you too see if they are normal or not as it really isn't too hard. However, as a short cut you should notice that once you have proved whether one of these 2-groups is normal or not you need not do it for the other ones because they are essentially the same, just with permuted symbols.

    For the quaternion group notice that <i> is a subgroup of order 4, and so the element i cannot be contained in any other proper subgroup of Q_8 (as the order of the subgroup must divide 8 and 4 is the largest number other than 8 which does that). The same goes for j and k. That gives us 3 proper subgroups. There is one more, which I shall let you find.

    I shall also leave you to find out whether these subgroups are normal or not - all you need to do to find out if the subgroup H is normal is to take every element h \in H and see if ghg^{-1} \in H holds for all g \in G. If it does H is normal, else it is not.

    Having types all that, can I ask you one thing: do you know of the result regarding normality and subgroups of index 2?

    Also, what does the title have to do with the content of the post - you don't do anything with homomorphisms here!
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