> Traditionally there are three political groups: conservatives, liberals and socialists.
That's not even approximately correct. There are a lot more groups than that, and those aren't even on the same conceptual level.
> "Far-left" falls under socialism, along with anarchism and communism.
Socialism is one of many leftist ideologies. Communism is, yes, a subset of socialism; OTOH, anarchism is not a subset of socialism (anarchism is a distinct ideology.)
> > The idea that ties them together is this concept of trying to fight against and eliminate some hierarchy in society. Often times, wealth inequality. There is no such thing in the US.
The first part is not horrendously inaccurate but not really useful correct, either.
I don't know if you are trying to say that the far left, socialism, anarchism, communism, social heirarchies, or wealth inequality doesn't exist in the US, but, in any case...no.
> Fascism falls under conservatism along with monarchism and any other traditionalist or reactionary political ideology.
Reactionary ideologies have the same relationship to conservatism as radical or far-left ideologies have to liberalism.
> Interesting to note that Republicanism is a liberal ideology, not a conservative one.
Small-r republicanism is an ideology that is neither inherently liberal nor conservative, though the most common modern form is liberal. Big-R Republicanism J's just attachment to an American political party that doesn't have a single clear ideology but covers a range of liberal to reactionary ideologies.
> Liberalism is a much more well defined ideology than the other two.
Socialism is, while still a fairly broad umbrella, both more specific and better defined than liberalism.
> Briefly speaking: "all men are created equal"
That doesn't actually differentiate it from socialism, nor is this really essential to liberalism (a belief that all men are created equal undergirds some liberalism, but isn't essential to liberalism; another basis is that equallegal treatment allows people to reach the different positions appropriate for different inherent worth.)
> (hence Republicanism)
Neither Republicanism nor republicanism is inherently attached to an egalitarian ideology.