Finding Cosets of Subgroups in Groups

[roam]

Homework Statement

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

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

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)

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 | \left\langle (4,3) \right\rangle | = |(4,3)|. 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?

 

[ystael]

Three notational problems:

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

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

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 o(g) for the order of an element g, which as you know equals the order |\langle g\rangle| 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 4 in U(5) = (\mathbb{Z}/(5))^\times is not 4, and the order of 3 in \mathbb{Z}_4 is not 3. Figure out what the correct orders are, and that should solve your problem.

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