# Finding the center of $\Bbb Z_5 \times\Bbb Z_{15}$

The question asks to find the center of $\mathbb{Z}_5 \times \mathbb{Z}_{15}$. I believe the answer is that the center is the whole group, since $\mathbb{Z}_5 \times \mathbb{Z}_{15} \cong \mathbb{Z}_5 \times \mathbb{Z}_5 \times \mathbb{Z}_3$, and since the group is of the form $\mathbb{Z}_{p_{1}^{i_1}} \times \cdots \times \mathbb{Z}_{p_{n}^{i_n}}$ where the $p_j$'s are primes (not necessarily distinct), then this must mean it is isomorphic to some Abelian group of order 75. Is this reasoning correct?

• The direct sum of two cyclic groups definitely is abelian and hence the center is the whole group. – Mathematician 42 Nov 28 '16 at 21:20