It might be worth explaining why you shouldn't expect $R(G)$ to tell you everything about a group. $R(G)$ is naturally isomorphic to the ring of class functions $G \to \mathbb{C}$ (the functions constant on conjugacy classes) under pointwise addition and multiplication, and as such the information it contains is precisely the multiset of sizes of each the number of conjugacy classes of $G$. That's it! No other information. (Note that $D_4$ and $Q$ both have conjugacy classes of sizes $1, 1, 2, 2, 2$ four five conjugacy classes.)
In other words, the abstract structure of the representation ring actually gives you less information than the character table; the character table at least hands you a distinguished basis of $R(G)$. Without this basis, $R(G)$ can't even tell you what tensor products of representations look like.
It might be worth explaining why you shouldn't expect $R(G)$ to tell you everything about a group. $R(G)$ is naturally isomorphic to the ring of class functions $G \to \mathbb{C}$ (the functions constant on conjugacy classes) under pointwise addition and multiplication, and as such the information it contains is precisely the multiset of sizes of each the number of conjugacy class classes of $G$. That's it! No other information. (Note that $D_4$ and $Q$ both have conjugacy classes of sizes $1, 1, 2, 2, 2$.2$four conjugacy classes.) In other words, the abstract structure of the representation ring actually gives you less information than the character table; the character table at least hands you a distinguished basis of$R(G)$. Without this basis,$R(G)$can't even tell you what tensor products of representations look like. 1 It might be worth explaining why you shouldn't expect$R(G)$to tell you everything about a group.$R(G)$is naturally isomorphic to the ring of class functions$G \to \mathbb{C}$(the functions constant on conjugacy classes) under pointwise addition and multiplication, and as such the information it contains is precisely the multiset of sizes of each conjugacy class of$G$. That's it! No other information. (Note that$D_4$and$Q$both have conjugacy classes of sizes$1, 1, 2, 2, 2$.) In other words, the abstract structure of the representation ring actually gives you less information than the character table; the character table at least hands you a distinguished basis of$R(G)$. Without this basis,$R(G)\$ can't even tell you what tensor products of representations look like.