Group homomorphisms and which of the following statements are true (NBHM-$2014$)

(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).

(a) is right. It is Cayley Theorem.

(b) is not right. For example, $|G|=p$ and if $H$ is a subgroup, $|H||(p-1)!$, so if $p$ is a prime, then $|G|\not=|H|$.

(c) is right. Consider the group ring $R(G)$ and you can define it as a linear space. Let $\pi:G\rightarrow{R(G)}$ satisfies $\pi(g):R(G)\rightarrow{R(G)}$:

$\pi(g)(\sum \sigma_i(g_i))=\sum \sigma_i(gg_i)$