Injective Homomorphism from a group into $GL_n$

Question

$|G|=n\ge 2<\infty,$ A group, I need to know which of the followings are true?

1. $\exists$ allways an injective homomorphism from $G$ into $S_n$

2. $\exists$ allways an injective homomorphism from $G$ into $S_m$ forsome $m<n$

3. $\exists$ allways an injective homomorphism from $G$ into $GL_n(\mathbb{R})$

$1$ is false, $2$ is true (Cayley's theorem), but I have no idea how to define such a map for $3$ if it exists. Thanks for helping.

Answer 1

Note that $2\Rightarrow 1$, so if 2 is true then so is 1. Also, $1\Rightarrow 3$ since the permutation matrices (the matrices that have all entries 0 except for exactly one 1 in each row and column) form a subgroup of the general linear group isomorphic to $S_n$. 1 is true by Cayley's theorem, but 2 is false as witnessed by the cyclic group of order 3.

As hinted by Hagen, the permutation matrices permute the elements of a basis for $\mathbb R^n$.

Answer 2

Your guesses for 1 and 2 are both wrong. For 3, try to find an injective homomorphism $S_n\to GL_n(\mathbb R)$. Hint: $S_n$ acts on an $n$-element set. What $n$-element thingy does an $n$-dimensional space have by definition?