Group Theory

• What can you say about finitely generated abelian groups?
• How many such groups are there of order 35, 27?
• What are the subgroups of these groups?
• Classify all abelian groups of 27 elements and prove that they are pairwise non-isomorphic (Bergman)
• Give an example of a non-abelian group of order 27. (Bergman)
• What can you say about groups of order pq?
• Show that any p-subgroup is contained in a p-Sylow subgroup.
• Show that any two p-Sylow subgroups are conjugate.
• Is the converse of Lagrange's theorem true?
• Give examples of a finite non-abelian group which cannot be mapped homomorphically onto a group of smaller order.
• Are the 2x2 matrices of determinant 1 simple?
• Can A₄ be mapped homomorphically onto ?
• Is the normalizer of a subgroup a normal subgroup?
• Show that G / H, where H is the commutator subgroup, is abelian.
• Show that is isomorphic to the rotations about the origin of the complex plane.
• Let G be the group of real numbers mod 1. What can be said about the subgroup generated by a single irrational number ?
• Is it true that for any finite group G there is a polynomial over such that G is the Galois group of the splitting field over of this polynomial?
• Let G be a group of order 99. Does there exist a normal subgroup of order 11 in G?
• Show that in a group of order 36, there is a nontrivial normal subgroup which has order either 3 or 9.
• Let H be a subgroup of G such that [G:H] is the smalles prime dividing the order of G. Show that H is normal.
• Show that any group of order 12 must have a normal subgroup.
• What is the order of the center of a nonabelian group of order p³, for a prime p?
• Let G be a group of order p^r for some prime p. Show that G has nontrivial center.
• Show that any group of order p^r, p a prime, is solvable.
• Show that if p divides the order of an abelian group, p a prime, then there is a subgroup of order p.
• If G is a cyclic group, and r divides the order of G, how many subgroups of order r does G have?
• Suppose that every element of a finite gropu G has order a power of the same prime. Is it true that the order of G is a power of this prime?
• Let S_p be a Sylow p-subgroup of G. Prove that N(N(S_p)) = N(S_p).
• Show that if | G | < 60 and | G | is not a prime, then G is not simple.
• If every subgroup of a group is normal, can you conclude that the group is abelian?
• What are the Sylow Theorems?
• What is a free group?
• What can you say about the subgroups of a group of order 30? (Stallings)
• Write down a composition series for S₄. (Bergman)
  • Which groups in this series are normal? (Bergman)
• Describe the 3-Sylow subgroups of symmetric groups. (Rhodes)
• Give an outline of the proof of the Jordan-Dickson Theorem. (Bergman)
• Describe using a picture the Sylow 3-subgroups of .
• Describe the 2-Sylow subgroups of S₄, S₅, and S₆. (Lam)
• Prove: If G is a group generated by 2 involutions, then G is a dihedral group. (Lam)
• State and prove the Core Theorem. (Bergman)
• How many (essentially different) 6 beaded necklaces using 2 colors are there? (Lam)
• Talk about multiple transitivity; what does it mean for a G-set to be k-transitive? Sharply k-transitive? (Bergman)
• Prove that for a sharply 2-transitive G-set X, the size of the Frobenius kernel is equal to the size of X. (Bergman)
• What groups are sharply k-transitive for large values of k? (Bergman)

This page was originally derived from this TeX file.