Why Are the 7 Sylow 2-Subgroups Intersection Trivial?

[math_grl]

I saw in an application of Sylow's theorems, it said we have something like a group of order 28 = 2^2 x 7, so we have either 1 or 7 sylow 2-subgroup. Assuming we have 7 sylow 2-subgroups, then we have 21 non-identity elements and the identity, and we have 1 sylow 7-subgroup, blah blah blah...

the point I wanted to know that was never really explained clearly was that why are the 7 sylow 2-subgroups intersection trivial? I realize that they are conjugates of each other, but this counting of elements seems kind of moot unless we know that these subgroups don't have any common elements.

 

[jbunniii]

Sylow subgroups do NOT in general have trivial intersection. What you can say, if you have 7 Sylow 2-subgroups, is that their union is at MOST 21 non-identity elements plus the identity.