I had a problem that goes like this: I have the grop G = (Z_{29}\{0}, multiplication). It has exactly one subgroup H with four elements and exactly one subgroup K with 14 elements. The task is to detemine H ∩ K - this group has a size ≠ 1.

We know the identity element is part of both, H and K wouldn't be groups else. Lagrange's theorem gives us that the size must be two.

Now, in the solution they do this to get the second element: the second element has an order that divides the number of elements in the group. Question #1: how do they know this? Is it perhaps a theorem that I've missed? Question #2: how can an element have an order? I know groups can have it (it's just the number of elements in the group), and permutations can have it (number of times you have to permute to get to start again), but a single element seems a bit counterintuitive.

From this they draw the conclusion that the element must solve the equation x^{2}- 1 = 0. Question #3: how do they get this equation? This "order" of the element were another number, saya, would I have to make an a:th degree polynomial, and how would I get the coefficients and other numbers? Besides, how do they get the - 1? Is that the identity element?

Grateful for answers!