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  1. #1
    Ife
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    group theory

    I am confused by the notation in this problem, i have an idea about showing a group to be a subgroup, but i am not sure what the notation means and i am not seeing it anywhere in my books or online, i am hoping someone can help.

    In G = {Z_{21}}^{x}, show that H = {[x]_{21}| x\equiv 1 (mod 3)} and K = {[x]_{21}| x\equiv 1 (mod 7)}
    are subgroups of G.
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  2. #2
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    Re: group theory

    first of all (\mathbb{Z}_{21})^\times is the group of units of the integers modulo 21 (all elements of \mathbb{Z}_{21} that have a multiplicative inverse with respect to multiplcation modulo 21).

    it turns out, that these are precisely the integers between 1 and 20 that are relatively prime to 21 (or more precisely, the equivalence classes of these integers mod 21).

    so (\mathbb{Z}_{21})^\times = \{1,2,4,5,8,10,11,13,16,17,19,20\}

    which of these numbers are congruent to 1 mod 3?

    clearly that set is {1,4,10,13,16,19} (note that this set has order 6, which divides 12).

    which of these numbers are congruent to 1 mod 7?

    these: {1,8}. note that this set also has order a divisor of 12.

    i'll show that {1,8} is a cyclic group of order 2, under multiplication mod 21. to do this, it suffices to show that 8 is of order 2:

    8^2 = 8*8 = 64 = 1 (mod 21) (since 64 = 63 + 1 = 21(3) + 1).

    if {1,4,10,13,16,19} is indeed a subgroup of (\mathbb{Z}_{21})^\times, then it is abelian (since multiplication mod 21 is commutatitive), and as such it must be cyclic of order 6. find a generator, and you're done!
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  3. #3
    Ife
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    Re: group theory

    Thank you so much for this, i was really unsure about the meaning of the notations. this helps a whole lot. because what i have seen in the texts thus far could have well been Greek!
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