Page 133 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
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mož Moravec: Some Topics in the Theory of Finite Groups 121
5. Prove that the group Cp Cpn is nilpotent of class precisely p n .
6. Let G be a group of order p n . If G has a unique subgroup of order p m for all 1 <
m < n, prove that G is cyclic.
7. Let G be a group of order p n , wher n ≥ 3, and of maximal class. Prove the following:
(a) G ab is an elementary abelian p -group of order p 2 and |γi (G ) : γi +1(G )| = p for
2 ≤ i ≤ n − 1. The group G can be generated by two elements.
(b) For every i ≥ 2 we have that γi (G ) is the only normal subgroup of G of index
pi.
(c) Zi (G ) = γn−i (G ) for all i = 0, . . . , n − 1.
8. Let G be a group in which x 2 ∈ Z (G ) for every x ∈ G . Prove the following:
(a) G is nilpotent of class ≤ 2.
(b) Every element of G has order at most 2.
(c) For all x , y ∈ G , the element (x y )2y −2x −2 belongs to G .
(d) For every x , y ∈ G we have that (x y )4 = x 4y 4.
9. Let G be a metabelian group and x , y , z , z 1, . . . , z n ∈ G . Prove:
(a) [x , y , z 1, . . . , z n ] = [x , y , z π(1), . . . , z π(n)] for every π ∈ Sn .
(b) [x , y , z ][y , z , x ][z , x , y ] = 1.
10. Let G be a group in which x 3 = 1 for all x ∈ G . Prove that [x , y , y ] = 1 for all x , y ∈ G .
11. Let G be a finite group and F its Fitting subgroup.
(a) Let N /F be an abelian normal subgroup of G /F such that N ≤ CG (F )F . Prove
that N = F (N ∩ CG (F )).
(b) Let N be as in (a). Prove that N /(N ∩ CG (F )) is nilpotent.
(c) Let c be the nilpotency class of N /(N ∩ CG (F )), where N is as above. Show
that N is nilpotent of class ≤ c + 1.
(d) Conclude that CG (F )F /F contains no non-trivial abelian normal subgroup.
(e) If G is solvable, show that CG (F ) ≤ F .
12. Let G be a finite nilpotent group and N a non-trivial normal subgroup of G . Show
the following:
(a) [N ,G ] is a proper subgroup of N .
(b) Some maximal proper subgroup of N is normal in G .
(c) Suppose that G is a p -group and M and N normal subgroups of G with N <
M . Prove that there exists K G such that N ≤ K < M and |M : K | = p .
5. Prove that the group Cp Cpn is nilpotent of class precisely p n .
6. Let G be a group of order p n . If G has a unique subgroup of order p m for all 1 <
m < n, prove that G is cyclic.
7. Let G be a group of order p n , wher n ≥ 3, and of maximal class. Prove the following:
(a) G ab is an elementary abelian p -group of order p 2 and |γi (G ) : γi +1(G )| = p for
2 ≤ i ≤ n − 1. The group G can be generated by two elements.
(b) For every i ≥ 2 we have that γi (G ) is the only normal subgroup of G of index
pi.
(c) Zi (G ) = γn−i (G ) for all i = 0, . . . , n − 1.
8. Let G be a group in which x 2 ∈ Z (G ) for every x ∈ G . Prove the following:
(a) G is nilpotent of class ≤ 2.
(b) Every element of G has order at most 2.
(c) For all x , y ∈ G , the element (x y )2y −2x −2 belongs to G .
(d) For every x , y ∈ G we have that (x y )4 = x 4y 4.
9. Let G be a metabelian group and x , y , z , z 1, . . . , z n ∈ G . Prove:
(a) [x , y , z 1, . . . , z n ] = [x , y , z π(1), . . . , z π(n)] for every π ∈ Sn .
(b) [x , y , z ][y , z , x ][z , x , y ] = 1.
10. Let G be a group in which x 3 = 1 for all x ∈ G . Prove that [x , y , y ] = 1 for all x , y ∈ G .
11. Let G be a finite group and F its Fitting subgroup.
(a) Let N /F be an abelian normal subgroup of G /F such that N ≤ CG (F )F . Prove
that N = F (N ∩ CG (F )).
(b) Let N be as in (a). Prove that N /(N ∩ CG (F )) is nilpotent.
(c) Let c be the nilpotency class of N /(N ∩ CG (F )), where N is as above. Show
that N is nilpotent of class ≤ c + 1.
(d) Conclude that CG (F )F /F contains no non-trivial abelian normal subgroup.
(e) If G is solvable, show that CG (F ) ≤ F .
12. Let G be a finite nilpotent group and N a non-trivial normal subgroup of G . Show
the following:
(a) [N ,G ] is a proper subgroup of N .
(b) Some maximal proper subgroup of N is normal in G .
(c) Suppose that G is a p -group and M and N normal subgroups of G with N <
M . Prove that there exists K G such that N ≤ K < M and |M : K | = p .
