Ciro Santilli @cirosantilli 37

By Ciro Santilli: Section "How to teach".

Others:

As per en.wikipedia.org/w/index.php?title=Semidirect_product&oldid=1040813965#Properties, unlike the Direct product, the semidirect product of two goups is neither unique, nor does it always exist, and there is no known algorithmic way way to tell if one exists or not.

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:

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

This is a general philosophy that Ciro Santilli, and likely others, observes over and over.

Basically, continuity, or higher order conditions like differentiability seem to impose greater constraints on problems, which make them more solvable.

Some good examples of that:

How are the bands measured experimentally?

Why are there gaps? Why aren't bands infinite? What determines the width of gaps?

Bibliography:

Inward Bound by Abraham Pais (1988) page 282 shows how this can be generalized from the Maxwell-Boltzmann distribution

Lots of demos.

Easily classified as the direct product of cyclic groups of prime order.

2021-08 bedroom battery out, 7.29V out of 9V duracell duralock. Buying pack of 12 Energizer nine-volt batteries. Measurement on new battery: 9.68V.

When I came, two Bell GU4 (MR11) 20W 12v.

One burnt. Put in an ASDA halogen one.

ASDA burnt, put in TopLux on right, old Bell left.

2019-01-24, right one burnt a few days ago, old Bell still works. Inner part black, and black dot on the wire. Putting new TopLux again, but this time on the left, old bell on right.

2019-01-24 toilet top lamp also burnt a few days ago, but not at the same time as mirror. Diall, 240V 40W, GU10. Putting in IKEA 240V 35W.

2019-02-02 toilet mirror lamp left (TopLux) burnt. Don't know what to do anymore. Only the magic Bell lamp works.

2019-03-06 toilet top lamp left burnt, IKEA 240V 35W GU10. Putting in another one.

2019-03-28 toilet top lamp right burnt, IKEA 240V 35W GU10. Waiting for people to come to look at transformer, there is definitely something wrong.

2019-04-03 top lamps: replaced with LED (LAP GU10 3W) since lower power, transformer not changed. Mirror lamps: transformer changed, left one replaced with Homebase Halogen 20W 12V. When I came back lamps flickering badly and sometimes not turning on, recalled technician.

2019-04-12 mirror lamp: it was just he connector that was bad, it was changed, also put LEDs there to make it less warm and hopefully have less tear on connector.