Genady

May I ask, in what senses they are not?

From Bug (engineering) - Wikipedia:

According to sources including wikipedia, it was a joke because the term had been in use already for a long time.

It was quite exciting and pleasant to watch our systems successfully crossing the date line in the time zone sequence first in Tokyo, then Singapore, London, and finally, NY. Being provided dinner and lodging in the downtown Manhattan, just in case.

You have shown that Since the answer to the question is Yes, you have shown that the alternating group has no subgroups of order half that of the group.

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. ?