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Homework Help: Order of Group

  1. Aug 21, 2010 #1
    1. The problem statement, all variables and given/known data

    [PLAIN]http://img641.imageshack.us/img641/8448/63794724.gif [Broken]

    3. The attempt at a solution

    Firstly, how do I list the elements of H?

    According to Lagrange's theorem if H is a subgroup of G then the number of distinct left (right) cosets of H in G is |G|\|H|.

    So I must find the orders of G and H:

    Since [tex]U(5)={1,2,3,4})[/tex] and [tex]\mathbb{Z}_4 = \{ 1,2,3,4 \})[/tex], the order of

    [tex]G=U(5) \oplus \mathbb{Z}_4 = (1,1),(1,2),(1,3),(1,4), (2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)[/tex]

    So G has order 16.

    H is generated by the element (4,3), where 4 is an element of U(5) and 3 is from Z4. I know that [tex]| \left\langle (4,3) \right\rangle | = |(4,3)|[/tex]. So I think

    |H|=|(4,3)|=lcm(|4|,|3|)=12

    Going back to lagrange's theorem |G|\|H|=16\12=4\3

    But how could the number of cosets be a fraction? Could anyone please show me what I did wrong?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Aug 24, 2010 #2
    Three notational problems:

    1. What is [tex]U(5)[/tex]? Is it the group [tex](\mathbb{Z}/(5))^\times[/tex], the multiplicative group of units of the integers modulo 5?

    2. If [tex]\mathbb{Z}_4[/tex] represents the additive group of integers modulo 4, it is conventional to choose representatives [tex]\{0, 1, 2, 3\}[/tex] rather than [tex]\{1, 2, 3, 4\}[/tex].

    3. The order of an element of a group (as opposed to the order of a group) is not written with absolute value bars. Some people write [tex]o(g)[/tex] for the order of an element [tex]g[/tex], which as you know equals the order [tex]|\langle g\rangle|[/tex] of the subgroup it generates. I think this notational confusion caused your mistakes below.

    Now, supposing I've guessed correctly about your notational issues, you have made two mistakes: the order of [tex]4[/tex] in [tex]U(5) = (\mathbb{Z}/(5))^\times[/tex] is not [tex]4[/tex], and the order of [tex]3[/tex] in [tex]\mathbb{Z}_4[/tex] is not [tex]3[/tex]. Figure out what the correct orders are, and that should solve your problem.

    You also haven't listed the elements of [tex]H = \langle (4, 3) \rangle[/tex], which the question asked for.
     
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