Ciro Santilli @cirosantilli  40

This is because reaching the "output" of the semidirect produt of two groups requires extra non-obvious information that might not exist. This is because the semi-direct product is based on the product of group subsets. So you start with two small and completely independent groups, and it is not obvious how to join them up, i.e. how to define the group operation of the product group that is compatible with that of the two smaller input groups. Contrast this with the Direct product, where the composition is simple: just use the group operation of each group on either side.

Product of group subsets

So in other words, it is not a function like the Direct product. The semidiret product is therefore more like a property of three groups.

The semidirect product is more general than the direct product of groups when thinking about the group extension problem, because with the direct product of groups, both subgroups of the larger group are necessarily also normal (trivial projection group homomorphism on either side), while for the semidirect product, only one of them does.

Conversely, en.wikipedia.org/w/index.php?title=Semidirect_product&oldid=1040813965 explains that if

G = N ⋊ H

, and besides the implied requirement that N is normal, H is also normal, then

G = N \times H

Smallest example:

D_{6} = C_{3} ⋊ C_{2}

where

D

is a dihedral group and

C

are cyclic groups.

C_{3}

(the rotation) is a normal subgroup of

D_{6}

, but

C_{2}

(the flip) is not.

Note that with the Direct product instead we get

C_{6}

and not

D_{6}

, i.e.

C_{3} \times C_{2} = C_{6}

as per the direct product of two cyclic groups of coprime order is another cyclic group.

TODO:

why does one of the groups have to be normal in the definition?
what is the smallest example of a non-simple group that is neither a direct nor a semi-direct product of any two other groups?

Bibliography: math.stackexchange.com/questions/1726939/is-this-intuition-for-the-semidirect-product-of-groups-correct

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Only people who need to drive a car should be allowed to drive a car anywhere near a city, e.g. people who work door to door, people who are disabled, etc.

Countryside driving is fine. If going to a city, you just have to drive to a parking outside of the city where you then take the public transport. And those who live in cities must leave their cars there too.

Everyone else must walk or cycle from home to public transport.

Cars just destroy everything, they make everything ugly:

this was extremely clear to Ciro Santilli as a cyclist. He previously lived in a place with few cars and the countryside was so pleasant. Then he moved to a place with more cars and it was shocking. It's a mixture of pollution, noise, and the fact that roads cut up the countryside that just make things not pleasant at all. Dual lane roads in particular are just a terrible thing. You can hear them from afar, much before you see them.
You can just see as tiny little villages surrounding the bit city and it's oversized motorways are more or less homogenized into one big city mass, the process is clearly visible as you cycle out of the big city and the villages become nicer and more unique as you go along further out.
even within cities, cars completely dehumanize the streets. For example, Ciro once lived in a small dead end street, and he would have gladly opened his front window more often to meet the neighbours. But just the noise of cars passing by every so often makes it impractical to work like that.

The Zatoichi effect applies well to the problem of cyclists:

they are not really pedestrians, and pedestrian paths are not suitable to them because they are too narrow, of not smooth, or curved. But pedestrians will always have enough political power to have their paths, because they live around the paths
they are not really motor vehicles, because motor vehicle paths are too wide and too fast for them. But motor vehicles will always have enough political power to have their paths, because people are lazy and stupid, and because as the world stands, individually you just don't have any reasonable choice to go anywhere.

This is the main drama faced by cyclists.