Product of two subgroups and intersection with p-subgroup

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SUMMARY

The discussion centers on the properties of Sylow p-subgroups within finite groups, specifically examining the intersection of a Sylow p-subgroup P with the product of a normal subgroup K and another subgroup H of G. It is established that if p divides |H|, then P ∩ HK is a subgroup of H. However, when K is not normal in G, the intersection may not retain subgroup properties, as illustrated by the counterexample S_3. The analysis confirms that the order of elements in the intersection is dictated by the orders of H and K.

PREREQUISITES
  • Understanding of Sylow theorems and their implications in group theory
  • Familiarity with subgroup properties and normal subgroups
  • Knowledge of group orders and their divisibility
  • Basic concepts of finite groups and their structure
NEXT STEPS
  • Study the implications of Sylow theorems in finite group classification
  • Explore the properties of normal subgroups and their significance in group actions
  • Investigate counterexamples in group theory, focusing on S_3 and its properties
  • Learn about the product of subgroups and conditions for subgroup formation
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This discussion is beneficial for mathematicians, particularly those specializing in group theory, as well as advanced students studying abstract algebra and its applications in finite group analysis.

moont14263
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Let G be a finite group. P is a Sylow p-subgroup of G and K is normal in G also H is a subgroup of G with (|K|,|H|)=1.
1) If p divides |H| then P\capHK is a subgroup of H.
2) Is (1) when K is not normal in G.

This is my try of (1);
Let y be an element of P\capHK, --> |y| divides |HK|=|H|*|K|--> |y| divides |H|--> y is an element of H as the order of y does not divide the order of K. What I am saying is that the elements of p power order of the intersection come from the Sylow p-subgroups of H.

For (2);
I know that |HK|=|H|*|K| but HK may not be a subgroup of G. And in this case also the intersection may not be a subgroup of G.

Help,me please.
 
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I can see that S_3 is a counter example for (1) and so there is no need to check (2). Thank you very much.
 

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