Why Are the 7 Sylow 2-Subgroups Intersection Trivial?

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The discussion centers on the intersection of Sylow 2-subgroups in a group of order 28, specifically addressing why the intersection of the 7 Sylow 2-subgroups is trivial. It is established that these subgroups are conjugates of each other, leading to the conclusion that they cannot share any common non-identity elements. The group structure implies that with 7 Sylow 2-subgroups, the total number of non-identity elements is limited to 21, reinforcing the notion of trivial intersection among these subgroups.

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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.
 
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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.
 

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