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Mathematics LibreTexts

13.4: Exercises

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1

Find all of the abelian groups of order less than or equal to 40 up to isomorphism.

2

Find all of the abelian groups of order 200 up to isomorphism.

3

Find all of the abelian groups of order 720 up to isomorphism.

4

Find all of the composition series for each of the following groups.

  1. Z12
  2. Z48
  3. The quaternions, Q8
  4. D4
  5. S3×Z4
  6. S4
  7. Sn, n5
  8. Q

5

Show that the infinite direct product G=Z2×Z2× is not finitely generated.

6

Let G be an abelian group of order m. If n divides m, prove that G has a subgroup of order n.

7

A group G is a torsion group if every element of G has finite order. Prove that a finitely generated abelian torsion group must be finite.

8

Let G, H, and K be finitely generated abelian groups. Show that if G×HG×K, then HK. Give a counterexample to show that this cannot be true in general.

9

Let G and H be solvable groups. Show that G×H is also solvable.

10

If G has a composition (principal) series and if N is a proper normal subgroup of G, show there exists a composition (principal) series containing N.

11

Prove or disprove: Let N be a normal subgroup of G. If N and G/N have composition series, then G must also have a composition series.

12

Let N be a normal subgroup of G. If N and G/N are solvable groups, show that G is also a solvable group.

13

Prove that G is a solvable group if and only if G has a series of subgroups

G=PnPn1P1P0={e}

where Pi is normal in Pi+1 and the order of Pi+1/Pi is prime.

14

Let G be a solvable group. Prove that any subgroup of G is also solvable.

15

Let G be a solvable group and N a normal subgroup of G. Prove that G/N is solvable.

16

Prove that Dn is solvable for all integers n.

17

Suppose that G has a composition series. If N is a normal subgroup of G, show that N and G/N also have composition series.

18

Let G be a cyclic p-group with subgroups H and K. Prove that either H is contained in K or K is contained in H.

19

Suppose that G is a solvable group with order n2. Show that G contains a normal nontrivial abelian subgroup.

20

Recall that the commutator subgroup G of a group G is defined as the subgroup of G generated by elements of the form a1b1ab for a,bG. We can define a series of subgroups of G by G(0)=G, G(1)=G, and G(i+1)=(G(i)).

  1. Prove that G(i+1) is normal in (G(i)). The series of subgroups

    G(0)=GG(1)G(2)

    is called the derived series of G.

  2. Show that G is solvable if and only if G(n)={e} for some integer n.

21

Suppose that G is a solvable group with order n2. Show that G contains a normal nontrivial abelian factor group.

22. Zassenhaus Lemma

Let H and K be subgroups of a group G. Suppose also that H and K are normal subgroups of H and K respectively. Then

  1. H(HK) is a normal subgroup of H(HK).
  2. K(HK) is a normal subgroup of K(HK).
  3. H(HK)/H(HK)K(HK)/K(HK)(HK)/(HK)(HK).

23. Schreier's Theorem

Use the Zassenhaus Lemma to prove that two subnormal (normal) series of a group G have isomorphic refinements.

24

Use Schreier's Theorem to prove the Jordan-Hölder Theorem.


This page titled 13.4: Exercises is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Thomas W. Judson (Abstract Algebra: Theory and Applications) via source content that was edited to the style and standards of the LibreTexts platform.

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