Homomorphisms: exercises

Practice (31)

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

1. Prove that \(\det( AB) = \det(A) \det(B)\) for \(A, B \in GL_2( {\mathbb R} )\). This shows that the determinant is a homomorphism from \(GL_2( {\mathbb R} )\) to \({\mathbb R}^*\).

2. Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel?

   1. \(\phi : {\mathbb R}^\ast \rightarrow GL_2 ( {\mathbb R})\) defined by \[\begin{aligned}\end{aligned}\]

   2. \(\phi : {\mathbb R} \rightarrow GL_2 ( {\mathbb R})\) defined by \[\begin{aligned}\end{aligned}\]

   3. \(\phi : GL_2 ({\mathbb R}) \rightarrow {\mathbb R}\) defined by \[\begin{aligned}\end{aligned}\]

   4. \(\phi : GL_2 ( {\mathbb R}) \rightarrow {\mathbb R}^\ast\) defined by \[\begin{aligned}\end{aligned}\]

   5. \(\phi : {\mathbb M}_2( {\mathbb R}) \rightarrow {\mathbb R}\) defined by \[\begin{aligned}\end{aligned}\], where \({\mathbb M}_2( {\mathbb R})\) is the additive group of \(2 \times 2\) matrices with entries in \({\mathbb R}\).

   Αποκάλυψέ την.

   Hint:

   (a) is a homomorphism with kernel \(\{ 1 \}\); (c) is not a homomorphism.

3. Let \(A\) be an \(m \times n\) matrix. Show that matrix multiplication, \(x \mapsto Ax\), defines a homomorphism \(\phi : {\mathbb R}^n \rightarrow {\mathbb R}^m\).

4. Let \(\phi : {\mathbb Z} \rightarrow {\mathbb Z}\) be given by \(\phi(n) = 7n\). Prove that \(\phi\) is a group homomorphism. Find the kernel and the image of \(\phi\).

   Αποκάλυψέ την.

   Hint:

   Since \(\phi(m + n) = 7(m+n) = 7m + 7n = \phi(m) + \phi(n)\), \(\phi\) is a homomorphism.

5. Describe all of the homomorphisms from \({\mathbb Z}_{24}\) to \({\mathbb Z}_{18}\).

   Αποκάλυψέ την.

   Hint:

   For any homomorphism \(\phi : {\mathbb Z}_{24} \rightarrow {\mathbb Z}_{18}\), the kernel of \(\phi\) must be a subgroup of \({\mathbb Z}_{24}\) and the image of \(\phi\) must be a subgroup of \({\mathbb Z}_{18}\). Now use the fact that a generator must map to a generator.

6. Describe all of the homomorphisms from \({\mathbb Z}\) to \({\mathbb Z}_{12}\).

7. In the group \({\mathbb Z}_{24}\), let \(H = \langle 4 \rangle\) and \(N = \langle 6 \rangle\).

   1. List the elements in \(HN\) (we usually write \(H + N\) for these additive groups) and \(H \cap N\).

   2. List the cosets in \(HN/N\), showing the elements in each coset.

   3. List the cosets in \(H/(H \cap N)\), showing the elements in each coset.

   4. Give the correspondence between \(HN/N\) and \(H/(H \cap N)\) described in the proof of the Second Isomorphism Theorem.

8. If \(G\) is an abelian group and \(n \in {\mathbb N}\), show that \(\phi : G \rightarrow G\) defined by \(g \mapsto g^n\) is a group homomorphism.

9. If \(\phi : G \rightarrow H\) is a group homomorphism and \(G\) is abelian, prove that \(\phi(G)\) is also abelian.

   Αποκάλυψέ την.

   Hint:

   Let \(a, b \in G\). Then \(\phi(a) \phi(b) = \phi(ab) = \phi(ba) = \phi(b)\phi(a)\).

10. If \(\phi : G \rightarrow H\) is a group homomorphism and \(G\) is cyclic, prove that \(\phi(G)\) is also cyclic.

11. Show that a homomorphism defined on a cyclic group is completely determined by its action on the generator of the group.

12. If a group \(G\) has exactly one subgroup \(H\) of order \(k\), prove that \(H\) is normal in \(G\).

13. Prove or disprove: \({\mathbb Q} / {\mathbb Z} \cong {\mathbb Q}\).

14. Let \(G\) be a finite group and \(N\) a normal subgroup of \(G\). If \(H\) is a subgroup of \(G/N\), prove that \(\phi^{-1}(H)\) is a subgroup in \(G\) of order \(|H| \cdot |N|\), where \(\phi : G \rightarrow G/N\) is the canonical homomorphism.

15. Let \(G_1\) and \(G_2\) be groups, and let \(H_1\) and \(H_2\) be normal subgroups of \(G_1\) and \(G_2\) respectively. Let \(\phi : G_1 \rightarrow G_2\) be a homomorphism. Show that \(\phi\) induces a homomorphism \(\overline{\phi} : (G_1/H_1) \rightarrow (G_2/H_2)\) if \(\phi(H_1) \subset H_2\).

16. If \(H\) and \(K\) are normal subgroups of \(G\) and \(H \cap K = \{ e \}\), prove that \(G\) is isomorphic to a subgroup of \(G/H \times G/K\).

17. Let \(\phi : G_1 \rightarrow G_2\) be a surjective group homomorphism. Let \(H_1\) be a normal subgroup of \(G_1\) and suppose that \(\phi(H_1) = H_2\). Prove or disprove that \(G_1/H_1 \cong G_2/H_2\).

    Αποκάλυψέ την.

    Hint:

    Find a counterexample.

18. Let \(\phi : G \rightarrow H\) be a group homomorphism. Show that \(\phi\) is one-to-one if and only if \(\phi^{-1}(e) = \{ e \}\).

19. Given a homomorphism \(\phi :G \rightarrow H\) define a relation \(\sim\) on \(G\) by \(a \sim b\) if \(\phi(a) = \phi(b)\) for \(a, b \in G\). Show this relation is an equivalence relation and describe the equivalence classes.

20. Let \(\aut(G)\) be the set of all automorphisms of \(G\); that is, isomorphisms from \(G\) to itself. Prove this set forms a group and is a subgroup of the group of permutations of \(G\); that is, \(\aut(G) \leq S_G\).

21. An inner automorphism of \(G\), \[\begin{aligned}\end{aligned}\], is defined by the map \[\begin{aligned}\end{aligned}\], for \(g \in G\). Show that \(i_g \in \aut(G)\).

22. The set of all inner automorphisms is denoted by \(\inn(G)\). Show that \(\inn(G)\) is a subgroup of \(\aut(G)\).

23. Find an automorphism of a group \(G\) that is not an inner automorphism.

24. Let \(G\) be a group and \(i_g\) be an inner automorphism of \(G\), and define a map \[\begin{aligned}\end{aligned}\] by \[\begin{aligned}\end{aligned}\]. Prove that this map is a homomorphism with image \(\inn(G)\) and kernel \(Z(G)\). Use this result to conclude that \[\begin{aligned}\end{aligned}\].

25. Compute \(\aut(S_3)\) and \(\inn(S_3)\). Do the same thing for \(D_4\).

26. Find all of the homomorphisms \(\phi : {\mathbb Z} \rightarrow {\mathbb Z}\). What is \(\aut({\mathbb Z})\)?

27. Find all of the automorphisms of \({\mathbb Z}_8\). Prove that \(\aut({\mathbb Z}_8) \cong U(8)\).

28. For \(k \in {\mathbb Z}_n\), define a map \(\phi_k : {\mathbb Z}_n \rightarrow {\mathbb Z}_n\) by \(a \mapsto ka\). Prove that \(\phi_k\) is a homomorphism.

29. Prove that \(\phi_k\) is an isomorphism if and only if \(k\) is a generator of \({\mathbb Z}_n\).

30. Show that every automorphism of \({\mathbb Z}_n\) is of the form \(\phi_k\), where \(k\) is a generator of \({\mathbb Z}_n\).

31. Prove that \(\psi : U(n) \rightarrow \aut({\mathbb Z}_n)\) is an isomorphism, where \(\psi : k \mapsto \phi_k\).

Symbols used here

\det A,\ |A|

determinant
Scaling factor of area/volume under A; zero means singular.

x \in A,\ A \subseteq B

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

A \cup B,\ A \cap B,\ A \setminus B

union, intersection, difference
In either; in both; in A but not B.

\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.

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

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