Since o(g_1) and o(g_2) are odd, they don't have a power of 2 in their prime decomposition. That means there are only odd numbers in their prime decomposition. Therefore , since o(g_1).o(g_2)=gcd(o(g_1), o(g_2)).lcm(o(g_1),o(g_2)), the number lcm(o(g_1),o(g_2)), which is o(g_1*g_2), has only odd prime factors, and thus is odd. I haven't used the hypothesis that the order of G is 4n+2, so does the question have subquestions?