Genady

I think so.

Nice. So, you've proved that a simple group cannot have subgroups of order half that of the group. 😉 Because I know the answer.

... if it's a subgroup of a finite group G and contains exactly half the elements of G?

I'd rather file a complaint.

Perhaps. But I don't see this number in the text.

What do they mean by this?

It is also sufficient, right?

"It can be also generated by its other element" is a given.

G is a cyclic group of order n generated by an element g, G = {e, g, g2, ..., gn-1). It can be also generated by its other element h = gk, G = {e, h, h2, ..., hn-1}. How are the k and n related?

I think I'm missing something. If g1 = h1 g h1-1 and g2 = h2 g h2-1 then g1 ~ g and g2 ~ g, but it is not necessary that g1 ~ g2, which they should if this is an equivalence class. ?

As Baku native, I could be offended (I am not.)

Does it mean, g1 ~ g when for some h1, g1 = h1 g h1-1 ?

IOW, Let's take any h in H. Because of H being closed, all combinations h, hh, hhh, ... are in H. But because of H being finite, some combinations should repeat, say hm = hn for some m<n. Then hn = hmhn-m = hm and thus hn-m is identity, e (from G). e = hn-m is thus contained in H. Now, if n-m = 1, then e = h is inverse of itself. Otherwise, e = hhn-m-1, which makes hn-m-1 an inverse of h.

I think, that being finite and closed guarantees that it contains the identity and all the inverses.

H is a finite subset of a group G, and for each h1, h2 in H their combination h1h2 is also in H. Is H a group?

Plus, even without other equivalent elements, each element forms a class of itself.

OK then.

Have you noticed that you reply to a four years old post to a member who did not visit since September 2021?

Equivalence relation R on set X defines equivalence classes in X. Do they form a partition of X?

I have applied for my retirement benefits in November. My application is still in process. I understand that eventually they will pay retroactively whatever accumulates since November. Without any interest. Thus, the OP question.

^^^^ fact, but ^^^^ speculation.

They have, e.g., Invasive Cup Coral | Flower Garden Banks National Marine Sanctuary

When the first plane crashed, I was straight under the WTC. 500m was how far I walked away before the structures collapsed.

It sure did. I watched it from about 500 m.

Almost three years after the OP, here is the status. One of the main lionfish hunters got bends while hunting lionfish and left the island long ago together with her partner, the other lionfish hunting champion. Since COVID, lionfish hunting practically has stopped. It is not promoted anymore. The result: no effect whatsoever. The lionfish is there, there is no more and no less of it, as well as the rest of the reef. Evidently, the devastating effect that has been predicted, was in fact a myth.