2004 November
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a. Carefully state (without proof) the three Sylow theorems.
b. Prove that every group of order p2q, where p and q are primes with p<q and p does not divide q−1, is abelian.
c. Prove that every group of order 12 has a normal Sylow subgroup. -
a. Among finite groups, define nilpotent group, in terms of a particular kind of normal series. State two conditions on G which are equivalent to the condition that G is nilpotent.
b. Prove that the center Z(G) of a nilpotent group is non-trivial.
c. Give an example which shows that N◃G with both N and G/N nilpotent is not sufficient for G to be nilpotent. -
a. If K,L are normal subgroups of G prove that G/K∩L is isomorphic to a subgroup of G/K×G/L (the external direct product). What is the index of this subgroup in G/K×G/L, in terms of [G:K],[G:L] and [G:KL]?
b. Prove either direction of: If N◃G, then G is solvable if and only if both N and G/N are solvable. -
a. If H is a subgroup of G and [G:H]=n≥2, prove that there exists a homomorphism ρ from G into Sn, the group of all permutations of an n-element set. Show that the kernel of ρ is contained in H, and the image of ρ is a transitive subgroup of Sn.
b. If [G:H]=n and G is simple, then G is isomorphic to a subgroup of An, the subgroup of all even permutations.
c. Every group of order 23⋅32⋅112 is solvable. -
Let G be a finite group, and suppose the automorphism group of G, Aut(G), acts transitively on G∖1. That is, whenever x,y∈G∖1 there exists an automorphism α∈Aut(G) such that α(x)=y. Prove that G is an elementary abelian p-group for some prime p, by proving that:
a. All non-identity elements of G have order p, for some prime p.
b. Here Z(G)≠1, and the center of every group is a characteristic subgroup.
c. Thus G≠Z(G), i.e., G is abelian. -
Suppose N is a normal subgroup of G. CG(N) denotes g∈G∣g−1ng=n for each n∈N.
a. Prove that CG(N)◃G, and if CG(N)=1 then |G| divides |N|!.
b. If N is also cyclic, prove that G/CG(N) is abelian, and hence that G′≤CG(N), where G′ is the commutator subgroup of G.