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Normal Sylow satisfies transfer condition

Statement

Statement with symbols

Suppose H is a normal Sylow subgroup of a finite group G and K \le G is any subgroup. Then H \cap K is a normal Sylow subgroup of K.

Related facts

More on transfer condition for normal and Sylow subgroups

Intersecting a normal subgroup with a Sylow subgroup

  • Equivalence of definitions of Sylow subgroup of normal subgroup: This states that a subgroup that is the intersection of a Sylow subgroup and a normal subgroup is a Sylow subgroup of the normal subgroup. Conversely, a subgroup expressible as a Sylow subgroup of a normal subgroup is expressible as an intersection of the normal subgroup with a Sylow subgroup of the whole group.

Facts used

Proof

Given: A finite group G, a normal Sylow subgroup H, and a subgroup K \le G.

To prove: H \cap K is a normal Sylow subgroup of K.

Proof: By the second isomorphism theorem, H \cap K is normal in K, and we have:

K/(H \cap K) \cong HK/H.

Here, HK is a subgroup of G. Since G/H has order relatively prime to p, so does HK/H (by fact (2)). Thus, K/(H \cap K) has order relatively prime to p, so [K:H \cap K] is relatively prime to p. Thus, H \cap K is a normal p-subgroup of K with index relatively prime to p, and is thus a normal p-Sylow subgroup of K.

References

Textbook references

  • Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Page 147, Exercise 32, Section 4.5 (Sylow's theorem)