Hw6 Problem 3

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Problem 3

Show that $\mathbb{Z}_p$ has no proper subgroups if p is prime.

Solution

We know that the orders of the subgroups of $\mathbb{Z}_p$ are the divisor of p by the Theorem of Lagrange.
However, p is prime, whose divisor are 1 and p. So $\mathbb{Z}_p$ has no proper subgroups and only has trivial subgroup and improper subgroup.

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