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Theorem about pgroups, similar to 3rd Sylow Theorem 
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#1
Dec1510, 01:03 AM

P: 37

(a) Let G be a finite group that is divisible by by p^k, and suppose that H is a subgroup of G with order p^j, where j is less than or equal to k. Show that the number of subgroups of G of order p^k that contain H is congruent to 1 modulo p.
(b) Find an example of a finite group that has exactly p+1 Sylow psubgroups. I think that I should be using the 3rd Sylow Theorem (the number of Sylow psubgroups of G is congruent to 1 modulo p) to prove (a). Also maybe the fact that since H is a pgroup, it is contained in Sylow psubgroup by the 2nd Sylow Theorem, and any larger pgroup containing H is also contained in a Sylow psubgroup. Any help would be much appreciated : ) 


#2
Dec1610, 07:16 PM

P: 234

The fact that pgroups have normal subgroups of all orders might prove useful, too.



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