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Let G be a finite group, and suppose that H is normal subgroup of G. Show...

  1. Let G be a finite group, and suppose that H is normal subgroup of G.

    1. Show that, for every g ∈ G, the order of gH in G/H must divide the order of g in G.

    2. What is the order of the coset [4]42 + 〈[6]42〉 in Z42/〈[6]42〉?
    3. Find an example to show that the order of gH in G/H does not always determine the order of g in G. That is, find an example of a group G, and a normal subgroup H is normal subgroup of G, and elements g1,g2 ∈ G, such that g1H = g2H but g1 and g2 do not have the same order in G. Prove that your example has all of the required properties.
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