>>11064498Based thread
>>11065747You're learning on you own from a textbook I assume? For some reason there are literally no textbooks that provide and informal or intuitive explanation of what the direct product is. Tbqh I don't even remember the formal definition off the top of my head, but it's literally just like adding on another dimension to a vector space. So like if you have the real numbers, you can take their direct product with themselves (the real numbers): you thereby obtain a new structure, R^2 space.
So yeah, think of it intuitively like combining copies of a vector space to get a new vector space. However, it's worth noting that the direct product can of course apply to structures other than a vector space, so you can talk about the direct product of groups, rings, etc. Either way, the notation is the same, so if x is an element of G and y is an element of H, then (x, y) is an element of their direct product G*H.
