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Unit 12: Sylow’s Theorems

12.4 Review Questions Notes

1. What are the order of all Sylow p-subgroups where G has order 18, 24, 54 and 80?

2. Find all the Sylow 3-subgroups of S and show that they are all conjugate.
4
3. Show that every group of order 45 has a normal subgroup of order 9.
4. Let H be a Sylow p-subgroup of G. Prove that H ps the only Sylow p-subgroup of G
contained in N(H).
5. Prove that no group of order 96 is simple.
6. If H is normal subgroup of a finite group G and |H| = p for some prime p, show that H is
k
a contained in every Sylow p-subgroup of G.

Answers: Self Assessment

1. (b) 2. (c) 3. (c) 4. (a) 5. (d)

12.5 Further Readings

Books Dan Saracino: Abstract Algebra; A First Course.
Mitchell and Mitchell: An Introduction to Abstract Algebra.
John B. Fraleigh: An Introduction to Abstract Algebra (Relevant Portion).

Online links www.jmilne.org/math/CourseNotes/
www.math.niu.edu
www.maths.tcd.ie/
archives.math.utk.edu

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