Exercises 13.4 Exercises
1.
Find all of the abelian groups of order less than or equal to up to isomorphism.
2.
Find all of the abelian groups of order up to isomorphism.
3.
Find all of the abelian groups of order up to isomorphism.
4.
Find all of the composition series for each of the following groups.
The quaternions,
5.
Show that the infinite direct product is not finitely generated.
6.
Let be an abelian group of order If divides prove that has a subgroup of order
7.
A group is a torsion group if every element of has finite order. Prove that a finitely generated abelian torsion group must be finite.
8.
Let and be finitely generated abelian groups. Show that if then Give a counterexample to show that this cannot be true in general.
9.
Let and be solvable groups. Show that is also solvable.
10.
If has a composition (principal) series and if is a proper normal subgroup of show there exists a composition (principal) series containing
11.
Prove or disprove: Let be a normal subgroup of If and have composition series, then must also have a composition series.
12.
Let be a normal subgroup of If and are solvable groups, show that is also a solvable group.
13.
Prove that is a solvable group if and only if has a series of subgroups
where is normal in and the order of is prime.
14.
Let be a solvable group. Prove that any subgroup of is also solvable.
15.
Let be a solvable group and a normal subgroup of Prove that is solvable.
16.
Prove that is solvable for all integers
17.
Suppose that has a composition series. If is a normal subgroup of show that and also have composition series.
18.
Let be a cyclic -group with subgroups and Prove that either is contained in or is contained in
19.
Suppose that is a solvable group with order Show that contains a normal nontrivial abelian subgroup.
20.
Recall that the commutator subgroup of a group is defined as the subgroup of generated by elements of the form for We can define a series of subgroups of by and
-
Prove that is normal in The series of subgroups
is called the derived series of
Show that is solvable if and only if for some integer
21.
Suppose that is a solvable group with order Show that contains a normal nontrivial abelian factor group.
22. Zassenhaus Lemma.
Let and be subgroups of a group Suppose also that and are normal subgroups of and respectively. Then
is a normal subgroup of
is a normal subgroup of
23. Schreier's Theorem.
Use the Zassenhaus Lemma to prove that two subnormal (normal) series of a group have isomorphic refinements.
24.
Use Schreier's Theorem to prove the Jordan-HΓΆlder Theorem.