Show that the direct sum $V \oplus W$ is a representation of the finite group $G$. Given that $V, W $ are representations.
attempt: Suppose that $V, W$ are vector spaces. Then define $\rho_1: G → GL(V)$ and $\rho_2: G → GL(W).$
Then the direct sum map $\rho_1 \oplus \rho_2 : G → GL(V \oplus W)$ is given by $(\rho_1 \oplus \rho_2)(g)(v,w) = (\rho_1(g)v , \rho_2(g)w)$.
for all $g \in G, v\in V, w\in W$.
Show that direct sum is indeed a representation.
attempt: Suppose that $g_1,g_2 \in G, v\in V, w\in W$. Then $(\rho_1 \oplus \rho_2)(g_1g_2)(v,w) = (\rho_1(g_1g_2)v , \rho_2(g_1g_2)w) = ((\rho_1(g_1), \rho_2(g_2))v, (\rho_1(g_1), \rho_2(g_2))w)$
I dont know how to continue or if I am even setting up the problem correctly. I know I have to show that the representation is closed under the action of G. Can someone please help me? I would really appreciate it. Thank you
I think you got a bit confused with your notation. What you should have in the last expression is $(\rho_1(g_1)\rho_1(g_2)v, \rho_2(g_1)\rho_2(g_2)w)$. Do you see how to proceed? If the $(\cdot, \cdot)$ notation is confusing, consider keeping the direct sum notation: $(\rho_1(g_1g_2)v)\oplus (\rho_2(g_1g_2)w)$ and so forth.