How to Clarify Sylow Subgroup Intersections in a Group of Order 48?

[Ad Infinitum NAU]

Hello all.. It's been quite some time since I've been here, so I doubt any of you remember me.

Anyhow, I'll get to my discussion..

I'm graduating in May with my BS in Math/Physics. I'm currently doing independent studies in Coding Theory as well as some higher abstract algebra.

I've been working on a problem I found in an old Abstract Algebra book for 3.5 weeks now and I finally have it solved but my details aren't clear enough for my satisfaction.

The detail I'm trying to pretty-up is: I've got three 2-Sylow subgroups of a group G where |G| = 48, and so the orders of the Hi's are 16 (where the Hi's are the 2-Sylow subgroups. I would like to show that |H1 intersect H2| = |H1 intersect H3| = |H2 intersect H3|

any idears?

 

[mathwonk]

aren't all the sylow subgroups conjugate to each other? does that help?

 

[Ad Infinitum NAU]

Yes, they are.. I've worked up to getting a homomorphism where: there is a g in G such that g*(H1 int H2)g^(-1) = H2 int H3, g*(H1 int H3)g^(-1) = H1 int H2, and g*(H2 int H3)g^(-1) = H1 int H3...

 

[mathwonk]

well? does that do it?

 

[Ad Infinitum NAU]

I can't recall all the details of an automorphism.. does it preserve order?

 

[mathwonk]

well an automorphism is a bijection that also preserves the group operation. you should be able to capitalize just on the fact it is a bijection.

 

[Ad Infinitum NAU]

I'm an idiot.. ha.. I love it when it's staring you right in the face like that haha.. thanks wonk.

 

[mathwonk]

it is indeed a pleasure to be of service.