(adsbygoogle = window.adsbygoogle || []).push({}); 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 Z_{4}. 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?

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# Homework Help: Order of Group

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