The Structure of Groups: exercises

Practice (25)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

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

   Jawaabta muuji

   Hint:

   There are three possible groups.

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. \({\mathbb Z}_{12}\)

   2. \({\mathbb Z}_{48}\)

   3. The quaternions, \(Q_8\)

   4. \(D_4\)

   5. \(S_3 \times {\mathbb Z}_4\)

   6. \(S_4\)

   7. \(S_n\), \(n \geq 5\)

   8. \({\mathbb Q}\)

   Jawaabta muuji

   Hint:

   (a) \(\{ 0 \} \subset \langle 6 \rangle \subset \langle 3 \rangle \subset {\mathbb Z}_{12}\); (e) \(\{ (1) \} \times \{ 0 \} \subset \{ (1), (1 \, 2 \, 3), (1 \, 3 \, 2) \} \times \{ 0 \} \subset S_3 \times \{ 0 \} \subset S_3 \times \langle 2 \rangle\subset S_3 \times {\mathbb Z}_4\).

5. Show that the infinite direct product \(G = {\mathbb Z}_2 \times {\mathbb Z}_2 \times \cdots\) 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.

   Jawaabta muuji

   Hint:

   Use the Fundamental Theorem of Finitely Generated Abelian Groups.

8. Let \(G\), \(H\), and \(K\) be finitely generated abelian groups. Show that if \(G \times H \cong G \times K\), then \(H \cong K\). Give a counterexample to show that this cannot be true in general.

9. Let \(G\) and \(H\) be solvable groups. Show that \(G \times H\) 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.

    Jawaabta muuji

    Hint:

    If \(N\) and \(G/N\) are solvable, then they have solvable series \[\begin{aligned}N = N_n \supset N_{n - 1} \supset \cdots \supset N_1 \supset N_0 = \{ e \} \\ G/N = G_n/N \supset G_{n - 1}/N \supset \cdots G_1/N \supset G_0/N = \{ N \}\end{aligned}\].

13. Prove that \(G\) is a solvable group if and only if \(G\) has a series of subgroups \[\begin{aligned}\end{aligned}\] where \(P_i\) is normal in \(P_{i + 1}\) and the order of \(P_{i + 1} / P_i\) 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 \(D_n\) is solvable for all integers \(n\).

    Jawaabta muuji

    Hint:

    Use the fact that \(D_n\) has a cyclic subgroup of index \(2\).

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 \(n \geq 2\). 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 \(a^{-1} b ^{-1} ab\) for \(a, b \in G\). 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 \[\begin{aligned}\end{aligned}\] 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 \(n \geq 2\). Show that \(G\) contains a normal nontrivial abelian factor group.

    Jawaabta muuji

    Hint:

    \(G/G'\) is abelian.

22. 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. \(H^* ( H \cap K^*)\) is a normal subgroup of \(H^* ( H \cap K)\).

    2. \(K^* ( H^* \cap K)\) is a normal subgroup of \(K^* ( H \cap K)\).

    3. \(H^* ( H \cap K) / H^* ( H \cap K^*) \cong K^* ( H \cap K) / K^* ( H^* \cap K) \cong (H \cap K) / (H^* \cap K)(H \cap K^*)\).

23. 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.

25. Write a program that will compute all possible abelian groups of order \(n\). What is the largest \(n\) for which your program will work?

Symbols used here

x \in A,\ A \subseteq B

element of, subset
x belongs to A; every element of A is in B.

i

imaginary unit
i² = −1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

\blacksquare\ \text{or}\ \square

end of proof (halmos)
Marks the point where the statement has been established.

a \equiv b \pmod n

congruent modulo n
n divides a − b; a and b have the same remainder.

a \mid b,\ \gcd(a,b)

divides, greatest common divisor
b is a multiple of a; the largest number dividing both.

(G, \cdot),\ e,\ g^{-1}

group, identity, inverse
A set with an operation; the do-nothing element; the element that undoes g.

G \cong H,\ G / N

isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.

\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n

integers modulo n
The remainders 0…n−1 with clock arithmetic.

\operatorname{Hom}(A, B),\ f \circ g

arrows from A to B, composition
The set of morphisms; do g then f.

Questions people ask

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group — that is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

Ku day inaad ku

⌨ Type ✎ Draw

Su'aalkaaga xisaabta

Xalin

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.

In ka badan Abstract Algebra

GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula