Number 3 hasn't really been answered yet from what i can see. Short answer: They are neither Long answer: Prime- and composite numbers are defined to be positive, but -p for a prime p is still a prime element of Z. The reason we use positive prime numbers is because they form a complete set of representatives (i think it's called in english), which means any integer can be written as z=u*p_1*p_2*...*p_n where u is a unit in Z (1 or -1) and p_i is prime. This would still hold if we chose some or all primes to be represented by their negative counterpart. So -5 is a prime element, but it's not what we would normally call a prime number (a prime element in the positive set of representatives). The same kind of logic applies to the composites. If we expand from prime numbers to prime elements, then and negative composite number could be written as the same product of primes, with arbitrarily one of them being negative.

Regarding Option 2 in the question: · "A primitive permutation group of degree $n$ containing a cycle of prime length $2\le p \le n-3$ must be the alternating or symmetric group." · This is a famous theorem by Jordan from the 1870s. It took a century to obtain that "prime" can be relaxed to "prime power", and afterwards it took the classification of finite simple groups to find out that the word "prime" can be dropped altogether, see · Peter M. Neumann, Primitive Permutation Groups Containing a Cycle of Prime-Power Length, Bulletin of the London Mathematical Society 7 (3), 1975, 298–299, · G. Jones, Primitive permutation groups containing a cycle, Bulletin of the Australian Mathematical Society, 89(1), 2014, 159-165.