Advanced Algebra I
Homework 4 due on Oct. 27, 2006
(1) * Complete the uncompleted proof in the lecture.
(2) Every subgroup and every quotient of a nilpotent group is nilpo- tent.
(3) Let G be a subgroup of S7generated by (1234567) and (34)(26).
Show that |G| = 168.
*One can even show that G is simple.
(4) If N C G and N ∩ G0 = {e}, then N < Z(G).
(5) (a) If p, q are prime. then every group of order p2q is solvable.
(b) If p, q are prime with p < q, then every group of order pqn is solvable.
(6) (a) A finite group of order < 60 is non-simple.
(b) A finite group of order < 60 is solvable.
(7) (a) Given a solvable group G, there is a subgroup H < G such that [G : H] is prime.
(b) Moreover, if H is a maximal proper subgroup of a finite solvable group G, then [G : H] is a prime power.
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