Group Theory – Normal subgroup implies $ g ^ 2 in H $ for each element $ g $

This question is Task 2.16 of An introduction to the theory of groups from Rotman.

2.16 If $ H le G $ has index $ 2 $, then $ g ^ 2 in H $ for each $ g in G $,

I know that the index is $ 2 $ implies that $ H $ is a normal subgroup. However, I do not fully understand how this helps to solve the problem. In addition, this exercise is performed before normal subgroups are even introduced, so it may be a motivating issue for normal subgroups, or may not even require that fact.

I know that after Lagrange's theorem:

$$ | G | = | H | cdot | G: H | $$
$$ | G | = 2 | H | $$

and so the subgroup is necessarily exactly half the size $ G $, This, too, clearly implies that $ G $ has an even number of elements, so I can make a statement about the parity of elements of odd and even order. However, I do not consider such a statement useful.

How should the statement be proved? The only guess I have at this point is that it has to do with the fact that cosets partition the group and that this, in combination with the above result, might possibly give the statement about Lagrange's theorem. Thoughts?

Elementary Theory – How do I refer to the only element in a singleton sentence?

You can rotate every element $ n $ into a singleton phrase by adding curly braces, $ {n } $, Is there an inverse to this operation, so if I know that the set is a singleton set, I can easily reference its element?

Suppose I have the set $ A = {5,6,7,8 } $and then a process that iteratively removes all elements except for one element of $ A $, and then I want to see what 10 plus is the resulting element, how could I write that?

The only way I can think of is to reference the first element by its index, e.g. $ 10 + A_1 $, Is there another way to do this? In my case the elements of $ A $ If there are already indices, I would have to index eg an index $ 10 + x_ {A_1} $, Which is not the worst thing in the world, I just wondered if there is a better way.

how do I get mpd url to download it via youtube-dl? I'm getting M4S URL from the Network tab in Inspect Element

How do I get a mpd-url of a video to download via youtube-dl?
In the "Network" tab, I get a URL like "m4s". "

Selection – HTML element: Quotation marks not selectable?

To include a quote in HTML, you could simply use quotation marks:

“Yes,” he said.

Alternatively, one could use the inline quotation mark:

Yes, he said.

The use of the inline quotation mark has some advantages, e.g. For example, it provides additional semantic information to any person or machine reading the HTML code.

However, I noticed something that seems to me to be a serious disadvantage of the inline offer element. In all browsers I've tried that, even though quotes are rendered, that's not possible choose the quotes.
Screenshot of unselected quotes

In Chrome and Edge, this predictably means that the quotation marks are omitted when the user copies and pastes. Interestingly, in Firefox, quotes are inserted into the inserted text even though they do not appear to be selected.

This behavior seems irritating to the user. Is it really the best way to quote in HTML? When should developers use the inline offer item?

dnd 5e – RAW, can a knight with swords and boards of the third level Absorb element make sense?

I often see Absorb Element (AE) as a useful spell for an Eldritch Knight (EK) because it gives you temporary resistance to a type of damage, and you can add that kind of damage to your next melee attack.

However, it does have somatic components that require a free hand to perform. For many spells, this is not a problem, as you can drop your weapon or stow it, cast it, and then use your bonus action for the EK Weapon Bond feature to resume it.
But since AE is a reaction, that does not seem to work, though I could be wrong. So it looks like you need to preemptively stash your weapon at the end of your turn if you think you need to use AE, which will be a problem if you actually end up taking a casual attack.

Of course, this all gets lost on the fourth level, if you can use War Caster as a feat that allows you to conjure up somatic components without a free hand.


  • Am I wrong? Is it possible to drop your sword as part of your reaction before acting on AE?
  • If not, can a third-level EK suitably use AE, or should you wait until the 4th level to use that spell?

java – Live element search on the phone

You can use coordinates to "hit" the click wherever you want, based on references that match the screen size.

Then screen with largura X e altura Y. My mouse goes to the cord 20x 10y and click.

Another option would be to use the Keyboard tab. Then you would count how many tabs are required to get to the item you want to click, and then click and enter as needed. and so on.

These are the two options you could use in these cases. I hope I helped.

Create a single AceGen element using the same constitutive model defined for two fiber families

I am currently working on a single constitutive model describing two collagen fiber families with different elemental properties. The strain energy density function is W, where there would be two values ​​of W for the two fiber families:

    `Wi = 1/2*c*(I1 - 3) + (k1/(2*k2) (Exp(k2*Ei)) - 
    1), (* i = 1, 2, these are the two fibre families*)
    Ei = Tr(hi)) - 1 (* i = 1, 2, a structure strain invariant for the two families*)
    hi1=k*bmod +(1 - (3*k))*(TensorProduct(a1, a1));
    hi2 =k*bmod + (1 - (3*k))*(TensorProduct(a2,  a2));`
(*a1 and a2 are vectors that describe the mean orientations of the fibre families with respect to the reference direction*)

I have not outlined all the variables, but I think this is sufficient to describe the general idea of ​​what the constitutive model outlines. Using the above information, a single AceGen element must be generated that describes both fiber families for later analysis in AceFem. Since I need to create an element containing 2 strain energy density functions instead of one, I was wondering if it's possible to create more than one tangent and remainder matrix in a tangent and remainder subroutine. My current subroutine for an optical fiber family looks like this (I did not enter the full code, but I think the relevant parts are just the tangent and residual compilation:

    SMSStandardModule("Tangent and residual");

skipR = SMSLogical(SMSInteger(idata$$("SkipResidual")) == 1);
skipK = SMSLogical(SMSInteger(idata$$("SkipTangent")) == 1);

NoIp = SMSInteger(es$$("id", "NoIntPoints"));

SMSDo(Ig, 1, NoIp);

    Export velocity and acceleration;
    SMSExport({v(1), v(2), v(3), v(4), v(5), v(6)}, 
  Table(ed$$("ht", Ihg + i), {i, 6}));

wgp = SMSReal(es$$("IntPoints", 4, Ig));

    SMSDo(i, 1, Length(pe);
        Rg1 = fGauss wgp  SMSD(
    W + T - ((Rho)*u.bforces), pe, i, "Constant" -> a);
    (*This process would have to be repeated twice for the fibre families W1 and W2*)

        SMSIf(! skipR); (*assembly of residual vector*)
 p$$(i), "AddIn" -> True);

Tangent stiffness;
        SMSDo(j, If(SMSSymmetricTangent, i, 1), Length(pe);
            Kg = SMSD(Rg1, pe, j);

            SMSExport(Kg, s$$(i, j), "AddIn" -> True);


But as I mentioned earlier, I have to add the second equation for the second fiber family, and I'm not sure if it's possible to create more than one tangent and remainder matrix in a tangent and residual subroutine. I came up with the following options:

1) Create a loop that goes through the material properties of the two fiber families, but I was told by a project leader that this is not necessary, as I could only write down the two equations, but I wonder how that would work.

2) I also thought that I could create two separate AceGen elements, but that would require a kind of overlay that I think is not possible when inputting in AceFem for finite element analysis.

Ideas are of course welcome, thank you in advance.