Hi, can someone help me understand the following question ....
"Let G be a finite group. Prove that G is a p-group iff every element of G has order a power of p."
ok this was my attempt at the question:
Suppose G is a p-group
, where
Let g
Hence, every element of G has order a power of p.
Conversely, suppose that every element of G has order a power of p
But
divides
Hence, G is a p-group.
can someone look over this and tell me what mistakes i made if any please ?
one implication is obvious:
if G is a p-group, (so that |G| = p^{n}) then the only possible divisors of |G| are powers of p, and since |g| divides |G|, |g| = p^{k}, for some 0 ≤ k ≤ n.
the other implication is not so obvious: if for any g in G,|g| = p^{k}, for some non-negative integer k, we know that |G| = p^{n}m, for some positive integer m with gcd(p,m) = 1, and some positive integer n.
but it is not clear that we must have m = 1.
however, by Cauchy's theorem, if q ≠ p is a prime that divides m, we must have an element x of G of order q. but there are no such elements, since every element of G has order a power of p. therefore, there are no prime numbers dividing m, so m = 1.