My group is $DP_{3,5}$. The set of ordered pairs (a, b) with $a\in Z_{3}$ and b$\in$ $Z_{5}$ under component-wise addition:

(a, b) + (c, d) = (a + c, b + d)

Associative Axiom:

let (a, b), (c, d), (e, f) with a, c, e$\in$ $Z_{3}$ and b, d, f$\in$ $Z_{5}$

((a, b) + (c, d)) + (e, f) = (a + c, b + d) + (e, f) = (a + c + e, b + d + f)

(a, b) + ((c, d) + (e, f)) = (a, b) + (c + e, d + f) = (a + c + e, b + d + f)

So it is associative.

Identity Axiom:

There exists an element e = (0, 0)

e + (a, b) = (0, 0) + (a, b) = (0 + a, 0 + b) = (a, b)

So it is identity.

Inverse Axiom:

For each element (a, b), there exists an element (a, b)' = -(a, b).

-(a, b) + (a, b) = (-a, -b) + (a, b) = (-a + a, -b + b) = (0, 0) which is identity e.

So it is inverse.

From G1, G2 and G3, we know that the order of pairs is group.

Subgroup and Cosets of $DP_{3,5}$.

Let (a, b) = (0, 1) and (c, d) = (1, 2)

Then (a, b) + (c, d) = (0, 1) + (1, 2) = (1, 3)

(1, 3) is a subgroup of $DP_{3,5}$.

Here are some cosets following:

( ) + (1, 3) = (1, 3)

(1, 0) + (1, 3) = (2, 3)

(2, 0) + (1, 3) = (0, 3)