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

  • Thread starter Ad Infinitum NAU
  • Start date
In summary, the speaker has returned after a long absence and is currently studying Math/Physics. They have been working on a problem involving three 2-Sylow subgroups of a group G with |G| = 48, and are trying to prove that the intersection of these subgroups all have the same order. They have found a homomorphism and are considering using an automorphism to prove their solution. They receive help from another member and realize the solution was right in front of them.
  • #1
Ad Infinitum NAU
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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?
 
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  • #2
aren't all the sylow subgroups conjugate to each other? does that help?
 
  • #3
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...
 
  • #4
well? does that do it?
 
  • #5
I can't recall all the details of an automorphism.. does it preserve order?
 
  • #6
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.
 
  • #7
I'm an idiot.. ha.. I love it when it's staring you right in the face like that haha.. thanks wonk.
 
  • #8
it is indeed a pleasure to be of service.
 

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

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