Nan Zheng

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)

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