Isomorphisms: exercises

Practice (40)

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 \(\mathbb Z \cong n \mathbb Z\) for \(n \neq 0\).

   Lafunua jibu

   Hint:

   Every infinite cyclic group is isomorphic to \({\mathbb Z}\) by .

2. Prove that \({\mathbb C}^\ast\) is isomorphic to the subgroup of \(GL_2( {\mathbb R} )\) consisting of matrices of the form \[\begin{aligned}\end{aligned}\].

   Lafunua jibu

   Hint:

   Define \(\phi: {\mathbb C}^* \rightarrow GL_2( {\mathbb R})\) by \[\begin{aligned}\end{aligned}\].

3. Prove or disprove: \(U(8) \cong {\mathbb Z}_4\).

   Lafunua jibu

   Hint:

   False.

4. Prove that \(U(8)\) is isomorphic to the group of matrices \[\begin{aligned}\end{aligned}\].

5. Show that \(U(5)\) is isomorphic to \(U(10)\), but \(U(12)\) is not.

6. Show that the \(n\)th roots of unity are isomorphic to \({\mathbb Z}_n\).

   Lafunua jibu

   Hint:

   Define a map from \({\mathbb Z}_n\) into the \(n\)th roots of unity by \(k \mapsto \cis(2k\pi / n)\).

7. Show that any cyclic group of order \(n\) is isomorphic to \({\mathbb Z}_n\).

8. Prove that \({\mathbb Q}\) is not isomorphic to \({\mathbb Z}\).

   Lafunua jibu

   Hint:

   Assume that \({\mathbb Q}\) is cyclic and try to find a generator.

9. Let \(G = {\mathbb R} \setminus \{ -1 \}\) and define a binary operation on \(G\) by \[\begin{aligned}\end{aligned}\]. Prove that \(G\) is a group under this operation. Show that \((G, *)\) is isomorphic to the multiplicative group of nonzero real numbers.

10. Show that the matrices \[\begin{aligned}\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \quad \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \\ \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}\end{aligned}\] form a group. Find an isomorphism of \(G\) with a more familiar group of order \(6\).

11. Find five non-isomorphic groups of order \(8\).

    Lafunua jibu

    Hint:

    There are two nonabelian and three abelian groups that are not isomorphic.

12. Prove \(S_4\) is not isomorphic to \(D_{12}\).

13. Let \(\omega = \cis(2 \pi /n)\) be a primitive \(n\)th root of unity. Prove that the matrices \[\begin{aligned}\end{aligned}\] generate a multiplicative group isomorphic to \(D_n\).

14. Show that the set of all matrices of the form \[\begin{aligned}\end{aligned}\], is a group isomorphic to \(D_n\), where all entries in the matrix are in \({\mathbb Z}_n\).

15. List all of the elements of \({\mathbb Z}_4 \times {\mathbb Z}_2\).

16. Find the order of each of the following elements.

    1. \((3, 4)\) in \({\mathbb Z}_4 \times {\mathbb Z}_6\)

    2. \((6, 15, 4)\) in \({\mathbb Z}_{30} \times {\mathbb Z}_{45} \times {\mathbb Z}_{24}\)

    3. \((5, 10, 15)\) in \({\mathbb Z}_{25} \times {\mathbb Z}_{25} \times {\mathbb Z}_{25}\)

    4. \((8, 8, 8)\) in \({\mathbb Z}_{10} \times {\mathbb Z}_{24} \times {\mathbb Z}_{80}\)

    Lafunua jibu

    Hint:

    (a) \(12\); (c) \(5\).

17. Prove that \(D_4\) cannot be the internal direct product of two of its proper subgroups.

18. Prove that the subgroup of \({\mathbb Q}^\ast\) consisting of elements of the form \(2^m 3^n\) for \(m,n \in {\mathbb Z}\) is an internal direct product isomorphic to \({\mathbb Z} \times {\mathbb Z}\).

19. Prove that \(S_3 \times {\mathbb Z}_2\) is isomorphic to \(D_6\). Can you make a conjecture about \(D_{2n}\)? Prove your conjecture.

    Lafunua jibu

    Hint:

    Draw the picture.

20. Prove or disprove: Every abelian group of order divisible by \(3\) contains a subgroup of order \(3\).

    Lafunua jibu

    Hint:

    True.

21. Prove or disprove: Every nonabelian group of order divisible by 6 contains a subgroup of order \(6\).

22. Let \(G\) be a group of order \(20\). If \(G\) has subgroups \(H\) and \(K\) of orders \(4\) and \(5\) respectively such that \(hk = kh\) for all \(h \in H\) and \(k \in K\), prove that \(G\) is the internal direct product of \(H\) and \(K\).

23. Prove or disprove the following assertion. Let \(G\), \(H\), and \(K\) be groups. If \(G \times K \cong H \times K\), then \(G \cong H\).

24. Prove or disprove: There is a noncyclic abelian group of order \(51\).

25. Prove or disprove: There is a noncyclic abelian group of order \(52\).

    Lafunua jibu

    Hint:

    True.

26. Let \(\phi : G \rightarrow H\) be a group isomorphism. Show that \(\phi( x) = e_H\) if and only if \(x=e_G\), where \(e_G\) and \(e_H\) are the identities of \(G\) and \(H\), respectively.

27. Let \(G \cong H\). Show that if \(G\) is cyclic, then so is \(H\).

    Lafunua jibu

    Hint:

    Let \(a\) be a generator for \(G\). If \(\phi :G \rightarrow H\) is an isomorphism, show that \(\phi(a)\) is a generator for \(H\).

28. Prove that any group \(G\) of order \(p\), \(p\) prime, must be isomorphic to \({\mathbb Z}_p\).

29. Show that \(S_n\) is isomorphic to a subgroup of \(A_{n+2}\).

30. Prove that \(D_n\) is isomorphic to a subgroup of \(S_n\).

31. Let \(\phi : G_1 \rightarrow G_2\) and \(\psi : G_2 \rightarrow G_3\) be isomorphisms. Show that \(\phi^{-1}\) and \(\psi \circ \phi\) are both isomorphisms. Using these results, show that the isomorphism of groups determines an equivalence relation on the class of all groups.

32. Prove \(U(5) \cong {\mathbb Z}_4\). Can you generalize this result for \(U(p)\), where \(p\) is prime?

33. Write out the permutations associated with each element of \(S_3\) in the proof of Cayley's Theorem.

34. An automorphism of a group \(G\) is an isomorphism with itself. Prove that complex conjugation is an automorphism of the additive group of complex numbers; that is, show that the map \(\phi( a + bi ) = a - bi\) is an isomorphism from \({\mathbb C}\) to \({\mathbb C}\).

35. Prove that \(a + ib \mapsto a - ib\) is an automorphism of \({\mathbb C}^*\).

36. Prove that \(A \mapsto B^{-1}AB\) is an automorphism of \(SL_2({\mathbb R})\) for all \(B\) in \(GL_2({\mathbb R})\).

37. We will denote the set of all automorphisms of \(G\) by \(\aut(G)\). \(\aut(G)\) automorphism group of a group \(G\) Prove that \(\aut(G)\) is a subgroup of \(S_G\), the group of permutations of \(G\).

38. Find \(\aut( {\mathbb Z}_6)\).

    Lafunua jibu

    Hint:

    Any automorphism of \({\mathbb Z}_6\) must send \(1\) to another generator of \({\mathbb Z}_6\).

39. Find \(\aut( {\mathbb Z})\).

40. Find two nonisomorphic groups \(G\) and \(H\) such that \(\aut(G) \cong \aut(H)\).

Symbols used here

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

matrix
A rectangular array of numbers; a linear map.

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.

\neq

not equal
The two sides are different.

\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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⌨ Type ✎ Draw

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Utasuluhisha

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.

Mengi zaidi katika 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