Cyclic groups and abelian groups

Question

Why abelian group of order 30 is always cyclic? I know that in a group of order 30 either subgroup of order 3 or subgroup of order 5 is normal

Answer 1

That comes from the abelian representation theorem. Since $30=2\cdot 3\cdot 5$ , the unique possible torsion coefficients is $(30) $, that means that all Abelian groups of order $30$ are $\mathbb{Z}_{30}$ up to isomorphism

Answer 2

Since the group is abelian it's the direct product of its Sylow subgroups.

$G=C_2×C_3×C_5\cong C_{30}$.

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By the same argument, an abelian group of order $p_1p_2\cdots p_k$, a product of distinct primes is cyclic.

Note that the isomorphism on the right is guaranteed by the Chinese remainder theorem.

You could also use the structure theorem.

Answer 3

$G$ abelian group of order $30$ .

$2, 3,5$ are prime divisors of $30$ , hence (by Cauchy's theorem) $G$ has elements $a, b, c$ of orders $2, 3,5$ respectively.

Then $g=abc$ has order $30$ ,hence $G=\langle g\rangle$