TIFR Mathematics 2022 | Part B | Question: 9

Question

Answer whether the following statements are True or False.

For any finite cyclic group $G$, there exists a prime power $q$ such that $G$ is a subgroup of $\mathbb{F}_{q}^{\times}.$

Answer 1

True:

Let G be a finite cyclic of order $n$.  Take any prime $p$ so that $\gcd(n,p)=1$,  By Euler’s theorem, $$p^{\phi(n)}=1\; (\mod \;n)$$ In other words, $n$ divides $p^{\phi(n)}-1. $ We know that , for any prime $p$,  there exist a field of order $p^{\phi(n)}$. So take our field as $F=F_q$ where $q= p^{\phi(n)}$.  Also multiplicative group is cyclic of order $q-1=p^{\phi(n)}-1$ and  $n$ divides $q-1$, so there exist a cyclic subgoup $H$ of  order $n$ in $F_q^*$. Any two cyclic group of same order are isomorphic, $G$ is isomorphic to $H$.