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Groups: exercises

Groups: exercises — from Judson, Abstract Algebra: Theory and Applications.

Practice (40)

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

Find all \(x \in {\mathbb Z}\) satisfying each of the following equations.
1. \(3x \equiv 2 \pmod{7}\)
2. \(5x + 1 \equiv 13 \pmod{23}\)
3. \(5x + 1 \equiv 13 \pmod{26}\)
4. \(9x \equiv 3 \pmod{5}\)
5. \(5x \equiv 1 \pmod{6}\)
6. \(3x \equiv 1 \pmod{6}\)
i

Hint:
(a) \(3 + 7 \mathbb Z = \{ \ldots, -4, 3, 10, \ldots \}\); (c) \(18 + 26 \mathbb Z\); (e) \(5 + 6 \mathbb Z\).
Which of the following multiplication tables defined on the set \(G = \{ a, b, c, d \}\) form a group? Support your answer in each case.
1. \[\begin{aligned}\end{aligned}\]
2. \[\begin{aligned}\end{aligned}\]
3. \[\begin{aligned}\end{aligned}\]
4. \[\begin{aligned}\end{aligned}\]
i

Hint:
(a) Not a group; (c) a group.
Write out Cayley tables for groups formed by the symmetries of a rectangle and for \(({\mathbb Z}_4, +)\). How many elements are in each group? Are the groups the same? Why or why not?
Describe the symmetries of a rhombus and prove that the set of symmetries forms a group. Give Cayley tables for both the symmetries of a rectangle and the symmetries of a rhombus. Are the symmetries of a rectangle and those of a rhombus the same?
Describe the symmetries of a square and prove that the set of symmetries is a group. Give a Cayley table for the symmetries. How many ways can the vertices of a square be permuted? Is each permutation necessarily a symmetry of the square? The symmetry group of the square is denoted by \(D_4\).
Give a multiplication table for the group \(U(12)\).

i

Hint:
\[\begin{aligned}\end{aligned}\]
Let \(S = {\mathbb R} \setminus \{ -1 \}\) and define a binary operation on \(S\) by \(a \ast b = a + b + ab\). Prove that \((S, \ast)\) is an abelian group.
Give an example of two elements \(A\) and \(B\) in \(GL_2({\mathbb R})\) with \(AB \neq BA\).

i

Hint:
Pick two matrices. Almost any pair will work.
Prove that the product of two matrices in \(SL_2({\mathbb R})\) has determinant one.
Prove that the set of matrices of the form \[\begin{aligned}\end{aligned}\] is a group under matrix multiplication. This group, known as the Heisenberg group, is important in quantum physics. Matrix multiplication in the Heisenberg group is defined by \[\begin{aligned}\end{aligned}\].
Prove that \(\det(AB) = \det(A) \det(B)\) in \(GL_2({\mathbb R})\). Use this result to show that the binary operation in the group \(GL_2({\mathbb R})\) is closed; that is, if \(A\) and \(B\) are in \(GL_2({\mathbb R})\), then \(AB \in GL_2({\mathbb R})\).
Let \({\mathbb Z}_2^n = \{ (a_1, a_2, \ldots, a_n) : a_i \in {\mathbb Z}_2 \}\). Define a binary operation on \({\mathbb Z}_2^n\) by \[\begin{aligned}\end{aligned}\]. Prove that \({\mathbb Z}_2^n\) is a group under this operation. This group is important in algebraic coding theory.
Show that \({\mathbb R}^{\ast} = {\mathbb R} \setminus \{0 \}\) is a group under the operation of multiplication.
Given the groups \({\mathbb R}^{\ast}\) and \({\mathbb Z}\), let \(G = {\mathbb R}^{\ast} \times {\mathbb Z}\). Define a binary operation \(\circ\) on \(G\) by \((a,m) \circ (b,n) = (ab, m + n)\). Show that \(G\) is a group under this operation.
Prove or disprove that every group containing six elements is abelian.

i

Hint:
There is a nonabelian group containing six elements.
Give a specific example of some group \(G\) and elements \(g, h \in G\) where \((gh)^n \neq g^nh^n\).

i

Hint:
Look at the symmetry group of an equilateral triangle or a square.
Give an example of three different groups with eight elements. Why are the groups different?

i

Hint:
The are five different groups of order 8.
Show that there are \(n!\) permutations of a set containing \(n\) items.

i

Hint:
Let \[\begin{aligned}\end{aligned}\] be in \(S_n\). All of the \(a_i\)s must be distinct. There are \(n\) ways to choose \(a_1\), \(n - 1\) ways to choose \(a_2\), , two ways to choose \(a_{n - 1}\), and only one way to choose \(a_n\). Therefore, we can form \(\sigma\) in \(n(n - 1) \cdots 2 \cdot 1 = n!\) ways.
Show that \[\begin{aligned}\end{aligned}\] for all \(a \in {\mathbb Z}_n\).
Prove that there is a multiplicative identity for the integers modulo \(n\): \[\begin{aligned}\end{aligned}\].
For each \(a \in {\mathbb Z}_n\) find an element \(b \in {\mathbb Z}_n\) such that \[\begin{aligned}\end{aligned}\].
Show that addition and multiplication mod \(n\) are well defined operations. That is, show that the operations do not depend on the choice of the representative from the equivalence classes mod \(n\).
Show that addition and multiplication mod \(n\) are associative operations.
Show that multiplication distributes over addition modulo \(n\): \[\begin{aligned}\end{aligned}\].
Let \(a\) and \(b\) be elements in a group \(G\). Prove that \(ab^na^{-1} = (aba^{-1})^n\) for \(n \in \mathbb Z\).

i

Hint:
\[\begin{aligned}(aba^{-1})^n & = (aba^{-1})(aba^{-1}) \cdots (aba^{-1}) \\ & = ab(aa^{-1})b(aa^{-1})b \cdots b(aa^{-1})ba^{-1} \\ & = ab^na^{-1}\end{aligned}\].
Let \(U(n)\) be the group of units in \({\mathbb Z}_n\). If \(n \gt 2\), prove that there is an element \(k \in U(n)\) such that \(k^2 = 1\) and \(k \neq 1\).
Prove that the inverse of \(g _1 g_2 \cdots g_n\) is \(g_n^{-1} g_{n-1}^{-1} \cdots g_1^{-1}\).
Prove the remainder of : if \(G\) is a group and \(a, b \in G\), then the equation \(xa = b\) has a unique solution in \(G\).
Prove .
Prove the right and left cancellation laws for a group \(G\); that is, show that in the group \(G\), \(ba = ca\) implies \(b = c\) and \(ab = ac\) implies \(b = c\) for elements \(a, b, c \in G\).
Show that if \(a^2 = e\) for all elements \(a\) in a group \(G\), then \(G\) must be abelian.

i

Hint:
Since \(abab = (ab)^2 = e = a^2 b^2 = aabb\), we know that \(ba = ab\).
Show that if \(G\) is a finite group of even order, then there is an \(a \in G\) such that \(a\) is not the identity and \(a^2 = e\).
Let \(G\) be a group and suppose that \((ab)^2 = a^2b^2\) for all \(a\) and \(b\) in \(G\). Prove that \(G\) is an abelian group.
Find all the subgroups of \({\mathbb Z}_3 \times {\mathbb Z}_3\). Use this information to show that \({\mathbb Z}_3 \times {\mathbb Z}_3\) is not the same group as \({\mathbb Z}_9\). (See for a short description of the product of groups.)
Find all the subgroups of the symmetry group of an equilateral triangle.

i

Hint:
\(H_1 = \{ \identity \}\), \(H_2 = \{ \identity, \rho_1, \rho_2 \}\), \(H_3 = \{ \identity, \mu_1 \}\), \(H_4 = \{ \identity, \mu_2 \}\), \(H_5 = \{ \identity, \mu_3 \}\), \(S_3\).
Compute the subgroups of the symmetry group of a square.
Let \(H = \{2^k : k \in {\mathbb Z} \}\). Show that \(H\) is a subgroup of \({\mathbb Q}^*\).
Let \(n = 0, 1, 2, \ldots\) and \(n {\mathbb Z} = \{ nk : k \in {\mathbb Z} \}\). Prove that \(n {\mathbb Z}\) is a subgroup of \({\mathbb Z}\). Show that these subgroups are the only subgroups of \(\mathbb{Z}\).
Let \({\mathbb T} = \{ z \in {\mathbb C}^* : |z| =1 \}\). Prove that \({\mathbb T}\) is a subgroup of \({\mathbb C}^*\).
Let \(G\) consist of the \(2 \times 2\) matrices of the form \[\begin{aligned}\end{aligned}\], where \(\theta \in {\mathbb R}\). Prove that \(G\) is a subgroup of \(SL_2({\mathbb R})\).

Symbols used here

\det A,\ |A|

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

a \equiv b \pmod n

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

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.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\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 \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.

Kuri Gukoresha

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 Abstract Algebra

Groups Subgroups, cosets and Lagrange's theorem Cyclic groups and permutation groups Homomorphisms, normal subgroups and quotient groups Rings and fields Galois theory: why the quintic has no formula