Determine the Number of Elements of Order 3 in a Non-Cyclic Group of Order 57

Group Theory Problems and Solutions

Problem 628

Let $G$ be a group of order $57$. Assume that $G$ is not a cyclic group.

Then determine the number of elements in $G$ of order $3$.
 
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Proof.

Observe the prime factorization $57=3\cdot 19$.
Let $n_{19}$ be the number of Sylow $19$-subgroups of $G$.

By Sylow’s theorem, we know that
\[n_{19} \equiv 1 \pmod{19} \text{ and } n_{19} \mid 3.\] It follows that $n_{19}=1$.


Now, observe that if $g\in G$, then the order of $g$ is $1$, $3$, or $19$. Note that since $G$ is not a cyclic group, the order of $g$ cannot be $57$.
As there is exactly one Sylow $19$-subgroup $P$, any element that is not in $P$ must have order $3$.


Therefore, the number of elements of order $3$ is $57-19=38$.

Remark.

Note that there are $16$ elements of order $19$ and the identity element is the only element of order $1$.


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