3.5: Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
Find all x∈Z satisfying each of the following equations.
- 3x≡2(mod7)
- 5x+1≡13(mod23)
- 5x+1≡13(mod26)
- 9x≡3(mod5)
- 5x≡1(mod6)
- 3x≡1(mod6)
Which of the following multiplication tables defined on the set G={a,b,c,d} form a group? Support your answer in each case.
-
∘abcdaacdabbbcdccdabddabc
-
∘abcdaabcdbbadcccdabddcba
-
∘abcdaabcdbbcdaccdabddabc
-
∘abcdaabcdbbacdccbaddddbc
Write out Cayley tables for groups formed by the symmetries of a rectangle and for (Z4,+). 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 D4.
Give a multiplication table for the group U(12).
Let S=R∖{−1} and define a binary operation on S by a∗b=a+b+ab. Prove that (S,∗) is an abelian group.
Give an example of two elements A and B in GL2(R) with AB≠BA.
Prove that the product of two matrices in SL2(R) has determinant one.
Prove that the set of matrices of the form
(1xy01z001)
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
(1xy01z001)(1x′y′01z′001)=(1x+x′y+y′+xz′01z+z′001).
Prove that det in GL_2({\mathbb R})\text{.} 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})\text{,} then AB \in GL_2({\mathbb R})\text{.}
Let {\mathbb Z}_2^n = \{ (a_1, a_2, \ldots, a_n) : a_i \in {\mathbb Z}_2 \}\text{.} Define a binary operation on {\mathbb Z}_2^n by
(a_1, a_2, \ldots, a_n) + (b_1, b_2, \ldots, b_n) = (a_1 + b_1, a_2 + b_2, \ldots, a_n + b_n)\text{.} \nonumber
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}\text{,} let G = {\mathbb R}^{\ast} \times {\mathbb Z}\text{.} Define a binary operation \circ on G by (a,m) \circ (b,n) = (ab, m + n)\text{.} Show that G is a group under this operation.
Prove or disprove that every group containing six elements is abelian.
Give a specific example of some group G and elements g, h \in G where (gh)^n \neq g^nh^n\text{.}
Give an example of three different groups with eight elements. Why are the groups different?
Show that there are n! permutations of a set containing n items.
Show that
0 + a \equiv a + 0 \equiv a \pmod{ n } \nonumber
for all a \in {\mathbb Z}_n\text{.}
Prove that there is a multiplicative identity for the integers modulo n\text{:}
a \cdot 1 \equiv a \pmod{n}\text{.} \nonumber
For each a \in {\mathbb Z}_n find an element b \in {\mathbb Z}_n such that
a + b \equiv b + a \equiv 0 \pmod{ n}\text{.} \nonumber
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\text{.}
Show that addition and multiplication mod n are associative operations.
Show that multiplication distributes over addition modulo n\text{:}
a(b + c) \equiv ab + ac \pmod{n}\text{.} \nonumber
Let a and b be elements in a group G\text{.} Prove that ab^na^{-1} = (aba^{-1})^n for n \in \mathbb Z\text{.}
Let U(n) be the group of units in {\mathbb Z}_n\text{.} If n \gt 2\text{,} prove that there is an element k \in U(n) such that k^2 = 1 and k \neq 1\text{.}
Prove that the inverse of g _1 g_2 \cdots g_n is g_n^{-1} g_{n-1}^{-1} \cdots g_1^{-1}\text{.}
Prove the remainder of Proposition 3.21: if G is a group and a, b \in G\text{,} then the equation xa = b has a unique solution in G\text{.}
Prove Theorem 3.23.
Prove the right and left cancellation laws for a group G\text{;} that is, show that in the group G\text{,} ba = ca implies b = c and ab = ac implies b = c for elements a, b, c \in G\text{.}
Show that if a^2 = e for all elements a in a group G\text{,} then G must be abelian.
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\text{.}
Let G be a group and suppose that (ab)^2 = a^2b^2 for all a and b in G\text{.} Prove that G is an abelian group.
Find all the subgroups of {\mathbb Z}_3 \times {\mathbb Z}_3\text{.} Use this information to show that {\mathbb Z}_3 \times {\mathbb Z}_3 is not the same group as {\mathbb Z}_9\text{.} (See Example 3.28 for a short description of the product of groups.)
Find all the subgroups of the symmetry group of an equilateral triangle.
Compute the subgroups of the symmetry group of a square.
Let H = \{2^k : k \in {\mathbb Z} \}\text{.} Show that H is a subgroup of {\mathbb Q}^*\text{.}
Let n = 0, 1, 2, \ldots and n {\mathbb Z} = \{ nk : k \in {\mathbb Z} \}\text{.} Prove that n {\mathbb Z} is a subgroup of {\mathbb Z}\text{.} Show that these subgroups are the only subgroups of \mathbb{Z}\text{.}
Let {\mathbb T} = \{ z \in {\mathbb C}^* : |z| =1 \}\text{.} Prove that {\mathbb T} is a subgroup of {\mathbb C}^*\text{.}
Let G consist of the 2 \times 2 matrices of the form
\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}\text{,} \nonumber
where \theta \in {\mathbb R}\text{.} Prove that G is a subgroup of SL_2({\mathbb R})\text{.}
Prove that
G = \{ a + b \sqrt{2} : a, b \in {\mathbb Q} \text{ and } a \text{ and } b \text{ are not both zero} \} \nonumber
is a subgroup of {\mathbb R}^{\ast} under the group operation of multiplication.
Let G be the group of 2 \times 2 matrices under addition and
H = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a + d = 0 \right\}\text{.} \nonumber
Prove that H is a subgroup of G\text{.}
Prove or disprove: SL_2( {\mathbb Z} )\text{,} the set of 2 \times 2 matrices with integer entries and determinant one, is a subgroup of SL_2( {\mathbb R} )\text{.}
List the subgroups of the quaternion group, Q_8\text{.}
Prove that the intersection of two subgroups of a group G is also a subgroup of G\text{.}
Prove or disprove: If H and K are subgroups of a group G\text{,} then H \cup K is a subgroup of G\text{.}
Prove or disprove: If H and K are subgroups of a group G\text{,} then H K = \{hk : h \in H \text{ and } k \in K \} is a subgroup of G\text{.} What if G is abelian?
Let G be a group and g \in G\text{.} Show that
Z(G) = \{ x \in G : gx = xg \text{ for all } g \in G \} \nonumber
is a subgroup of G\text{.} This subgroup is called the center of G\text{.}
Let a and b be elements of a group G\text{.} If a^4 b = ba and a^3 = e\text{,} prove that ab = ba\text{.}
Give an example of an infinite group in which every nontrivial subgroup is infinite.
If xy = x^{-1} y^{-1} for all x and y in G\text{,} prove that G must be abelian.
Prove or disprove: Every proper subgroup of a nonabelian group is nonabelian.
Let H be a subgroup of G and
C(H) = \{ g \in G : gh = hg \text{ for all } h \in H \}\text{.} \nonumber
Prove C(H) is a subgroup of G\text{.} This subgroup is called the centralizer of H in G\text{.}
Let H be a subgroup of G\text{.} If g \in G\text{,} show that gHg^{-1} = \{ghg^{-1} : h\in H\} is also a subgroup of G\text{.}