So I have two questions $1)$ on multiplicative inverses, and $2)$ on group action.
$1)$ Why is the group action $GL(n,\mathbb{R})$ on to a set $\mathbb{R}^n$ faithful but not transitive?
$2)$ Why is the symmetric group $S_n$ acting on $X=\{1,2,3,\dots,n\}$ faithful and transitive?
This chapter on group actions really knocking my head out, I dont get what's happening at all, can anyone help me clarify the above examples?
Next I want to ask about multiplicative inverses. What are multiplicative inverses in some set $R$ exactly? For instance is it any set of $x$ that satisfies the inequality $ax= 1\bmod n$ ,for $x \in R$?