Statement#

Let $G$ be a group of prime order $p$, where $p$ is a prime number.

Prove that:

1. Every non-identity element of $G$ generates the entire group
2. Therefore, $G$ is cyclic

Required Topics#

• Lagrange's Theorem
• Cyclic groups
• Order of elements and subgroups
• Properties of prime numbers

Lagrange's Theorem#

For any finite group $G$ and subgroup $H \leq G$: $|H| \text{ divides } |G|$

What to Prove#

Show that if $g \in G$ and $g \neq e$ (identity), then $\langle g \rangle = G$.

Solution#

Solution coming soon.