TITLE: Group homomorphisms and which of the following statements are true (NBHM-$2014$)
QUESTION [5 upvotes]: Let $G$ be a finite group of order $n\ge2$. Which of the following statements are true?
a. There always exists an injective homomorphism from $G$ into $S_n$.
b. There always exists an injective homomorphism from $G$ into $S_m$ for some $m\lt n$.
c. There always exists an injective homomorphism from $G$ into $GL_n(R)$.

I think b. is right (don't know how to prove though). I also think that c. is false. But not sure.

REPLY [6 votes]: (a) is true by Cayley's theorem.
(b) is not true as $\mathbb{Z}_3$ is not a subgroup of $S_2$ and $S_1$. 
(c) is true as $S_n$ is isomorphic to the group of all permutation matrices inside $GL_n(\mathbb{R})$, so (a) implies (c).