Quick (and basic) group theory question:
Say G, H, K some (Lie) groups, does it in general hold that $$ (G \times H)/H = G $$ and that $$ H = K\times G \to K = H / G $$ And if so, does it then also hold that $$ (H/G)/K = (H/K)/G $$ where throughout for simplicity I assume every time I take a quotient that the group I quotient by is a normal subgroup of the group that is quotiented and I defined "=" here as up to an isomorphism. Thanks in advance!
Take homomorphism $G\times H \to G$ such that $(g,h)\to g$, then check it is onto and kernel is $H$ and use Isomorphism Theorem.
Second follows from first.
Third does not make sense unless we have subgroup relation among $H,G,K$ as you can define $G/H$ only if $H$ is normal subgroup of $G$