Some Properties of p-Groups and Commutative p-Groups

For simplicity, we use the following convention: G is a group, a, b are elements of G, m, n are natural numbers, and p is a prime natural number. One can prove the following propositions: (1) If for every natural number r holds n 6= pr, then there exists an element s of N such that s is prime and s | n and s 6= p. (2) For all natural numbers n, m such that n | pm there exists a natural number r such that n = pr and r ≤ m. (3) If an = 1G, then (a−1) n = 1G. (4) If (a−1) = 1G, then an = 1G. (5) ord(a−1) = ord(a). (6) ord(ab) = ord(a). (7) Let G be a group, N be a subgroup of G, and a, b be elements of G. Suppose N is normal and b ∈ N. Let given n. Then there exists an element g of G such that g ∈ N and (a · b) = an · g.