8.03c Group definition: recall and use, show structure is/isn't a group

11

1. (a) Given \(\mathbf { A } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } e & f \\ g & h \end{array} \right)\), work out the matrix \(\mathbf { A B }\) and write down expressions for \(\operatorname { det } \mathbf { A }\) and \(\operatorname { det } \mathbf { B }\).
   (b) Verify, by direct calculation, that \(\operatorname { det } ( \mathbf { A B } ) = \operatorname { det } \mathbf { A } \times \operatorname { det } \mathbf { B }\). Let \(S\) be the set of all \(2 \times 2\) matrices with determinant equal to 1 .
2. Show that \(\left( S , \times _ { \mathrm { M } } \right)\) forms a group, \(G\), where \(\times _ { \mathrm { M } }\) is the operation of matrix multiplication. [You may assume that \(\mathrm { X } _ { \mathrm { M } }\) is associative.]
3. (a) Show that \(\mathbf { K } = \left( \begin{array} { l l } 1 & \mathrm { i } \\ \mathrm { i } & 0 \end{array} \right)\) is an element of \(G\). Let \(H\) be the smallest subgroup of \(G\) that contains \(\mathbf { K }\) and let \(n\) be the order of \(H\).
   (b) Determine the value of \(n\).
   (c) Give a second subgroup of \(G\), also of order \(n\), which is isomorphic to \(H\).

8 Consider the set \(S\) of all matrices of the form \(\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)\), where p is a non-zero rational number.

1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). (You may assume that matrix multiplication is associative.)
2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3 .

8 Consider the set \(S\) of all matrices of the form \(\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)\), where p is a non-zero rational number.

1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). (You may assume that matrix multiplication is associative.)
2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3.

6 The set \(S\) consists of all real numbers except 1. The binary operation * is defined for all \(a , b\) in \(S\) by $$a * b = a + b - a b$$

1. By considering the identity \(a + b - a b \equiv 1 - ( a - 1 ) ( b - 1 )\), or otherwise, show that \(S\) is closed under *.
2. Show that * is associative on \(S\).
3. Find the identity of \(S\) under \(*\), and the inverse of \(x\) for all \(x \in S\).
4. The set \(S\), together with the binary operation *, forms a group \(G\). Find a subgroup of \(G\) of order 2 .

10 A group \(G\) has distinct elements \(e , a , b , c , \ldots\), where \(e\) is the identity element and \(\circ\) is the binary operation.

1. Prove that if \(a \circ a = b\) and \(b \circ b = a\), then the set of elements \(\{ e , a , b \}\) forms a subgroup of \(G\).
2. Prove that if \(a \circ a = b , b \circ b = c\) and \(c \circ c = a\), then the set of elements \(\{ e , a , b , c \}\) does not form a subgroup of \(G\).

Elements of the set \(\{p, q, r, s, t\}\) are combined according to the operation table shown below.

	\(p\)	\(q\)	\(r\)	\(s\)	\(t\)
\(p\)	\(t\)	\(s\)	\(p\)	\(r\)	\(q\)
\(q\)	\(s\)	\(p\)	\(q\)	\(t\)	\(r\)
\(r\)	\(p\)	\(q\)	\(r\)	\(s\)	\(t\)
\(s\)	\(r\)	\(t\)	\(s\)	\(q\)	\(p\)
\(t\)	\(q\)	\(r\)	\(t\)	\(p\)	\(s\)

1. Verify that \(q(st) = (qs)t\). [2]
2. Assuming that the associative property holds for all elements, prove that the set \(\{p, q, r, s, t\}\), with the operation table shown, forms a group \(G\). [4]
3. A multiplicative group \(H\) is isomorphic to the group \(G\). The identity element of \(H\) is \(e\) and another element is \(d\). Write down the elements of \(H\) in terms of \(e\) and \(d\). [2]

The set \(S\) consists of the numbers \(3^n\), where \(n \in \mathbb{Z}\). (\(\mathbb{Z}\) denotes the set of integers \(\{0, \pm 1, \pm 2, \ldots\}\).)

1. Prove that the elements of \(S\), under multiplication, form a commutative group \(G\). (You may assume that addition of integers is associative and commutative.) [6]
2. Determine whether or not each of the following subsets of \(S\), under multiplication, forms a subgroup of \(G\), justifying your answers.
   1. The numbers \(3^{2n}\), where \(n \in \mathbb{Z}\). [2]
   2. The numbers \(3^n\), where \(n \in \mathbb{Z}\) and \(n \geqslant 0\). [2]
   3. The numbers \(3^{(±n^2)}\), where \(n \in \mathbb{Z}\). [2]

Groups \(A, B, C\) and \(D\) are defined as follows: \begin{align} A: &\quad \text{the set of numbers } \{2, 4, 6, 8\} \text{ under multiplication modulo 10,}
B: &\quad \text{the set of numbers } \{1, 5, 7, 11\} \text{ under multiplication modulo 12,}
C: &\quad \text{the set of numbers } \{2^0, 2^1, 2^2, 2^3\} \text{ under multiplication modulo 15,}
D: &\quad \text{the set of numbers } \left\{\frac{1+2m}{1+2n}, \text{ where } m \text{ and } n \text{ are integers}\right\} \text{ under multiplication.} \end{align}

1. Write down the identity element for each of groups \(A, B, C\) and \(D\). [2]
2. Determine in each case whether the groups \begin{align} &A \text{ and } B,
   &B \text{ and } C,
   &A \text{ and } C \end{align} are isomorphic or non-isomorphic. Give sufficient reasons for your answers. [5]
3. Prove the closure property for group \(D\). [4]
4. Elements of the set \(\left\{\frac{1+2m}{1+2n}, \text{ where } m \text{ and } n \text{ are integers}\right\}\) are combined under addition. State which of the four basic group properties are not satisfied. (Justification is not required.) [2]

It is given that the set of complex numbers of the form \(re^{i\theta}\) for \(-\pi < \theta \leqslant \pi\) and \(r > 0\), under multiplication, forms a group.

1. Write down the inverse of \(5e^{3\pi i}\). [1]
2. Prove the closure property for the group. [2]
3. \(Z\) denotes the element \(e^{i\gamma}\), where \(\frac{1}{2}\pi < \gamma < \pi\). Express \(Z^2\) in the form \(e^{i\theta}\), where \(-\pi < \theta \leqslant 0\). [2]

Groups \(A\), \(B\), \(C\) and \(D\) are defined as follows: \(A\): the set of numbers \(\{2, 4, 6, 8\}\) under multiplication modulo 10, \(B\): the set of numbers \(\{1, 5, 7, 11\}\) under multiplication modulo 12, \(C\): the set of numbers \(\{2^0, 2^1, 2^2, 2^3\}\) under multiplication modulo 15, \(D\): the set of numbers \(\left\{\frac{1+2m}{1+2n}, \text{ where } m \text{ and } n \text{ are integers}\right\}\) under multiplication.

1. Write down the identity element for each of groups \(A\), \(B\), \(C\) and \(D\). [2]
2. Determine in each case whether the groups
   \(A\) and \(B\), \(B\) and \(C\), \(A\) and \(C\)
   are isomorphic or non-isomorphic. Give sufficient reasons for your answers. [5]
3. Prove the closure property for group \(D\). [4]
4. Elements of the set \(\left\{\frac{1+2m}{1+2n}, \text{ where } m \text{ and } n \text{ are integers}\right\}\) are combined under addition. State which of the four basic group properties are not satisfied. (Justification is not required.) [2]

\(H\) denotes the set of numbers of the form \(a + b\sqrt{5}\), where \(a\) and \(b\) are rational. The numbers are combined under multiplication.

1. Show that the product of any two members of \(H\) is a member of \(H\). [2] It is now given that, for \(a\) and \(b\) not both zero, \(H\) forms a group under multiplication.
2. State the identity element of the group. [1]
3. Find the inverse of \(a + b\sqrt{5}\). [2]
4. With reference to your answer to part (iii), state a property of the number 5 which ensures that every number in the group has an inverse. [1]

The function f is defined by \(\text{f} : x \mapsto \frac{1}{2-2x}\) for \(x \in \mathbb{R}, x \neq 0, x \neq \frac{1}{2}, x \neq 1\). The function g is defined by \(\text{g}(x) = \text{ff}(x)\).

1. Show that \(\text{g}(x) = \frac{1-x}{1-2x}\) and that \(\text{gg}(x) = x\). [4]

It is given that f and g are elements of a group \(K\) under the operation of composition of functions. The element e is the identity, where \(\text{e} : x \mapsto x\) for \(x \in \mathbb{R}, x \neq 0, x \neq \frac{1}{2}, x \neq 1\).

1. State the orders of the elements f and g. [2]
2. The inverse of the element f is denoted by h. Find \(\text{h}(x)\). [2]
3. Construct the operation table for the elements e, f, g, h of the group \(K\). [4]

The operation \(*\) is defined on the elements \((x, y)\), where \(x, y \in \mathbb{R}\), by $$(a, b) * (c, d) = (ac, ad + b).$$ It is given that the identity element is \((1, 0)\).

1. Prove that \(*\) is associative. [3]
2. Find all the elements which commute with \((1, 1)\). [3]
3. It is given that the particular element \((m, n)\) has an inverse denoted by \((p, q)\), where $$(m, n) * (p, q) = (p, q) * (m, n) = (1, 0).$$ Find \((p, q)\) in terms of \(m\) and \(n\). [2]
4. Find all self-inverse elements. [3]
5. Give a reason why the elements \((x, y)\), under the operation \(*\), do not form a group. [1]

A group \(D\) of order 10 is generated by the elements \(a\) and \(r\), with the properties \(a^2 = e\), \(r^5 = e\) and \(r^4a = ar\), where \(e\) is the identity. Part of the operation table is shown below. \includegraphics{figure_1}

1. Give a reason why \(D\) is not commutative. [1]
2. Write down the orders of any possible proper subgroups of \(D\). [2]
3. List the elements of a proper subgroup which contains
   1. the element \(a\), [1]
   2. the element \(r\). [1]
4. Determine the order of each of the elements \(r^3\), \(ar\) and \(ar^2\). [4]
5. Copy and complete the section of the table marked E, showing the products of the elements \(ar\), \(ar^2\), \(ar^3\) and \(ar^4\). [5]

A multiplicative group with identity \(e\) contains distinct elements \(a\) and \(r\), with the properties \(r^6 = e\) and \(ar = r^2a\).

1. Prove that \(rar = a\). [2]
2. Prove, by induction or otherwise, that \(r^n ar^n = a\) for all positive integers \(n\). [4]

A set of matrices \(M\) is defined by $$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} \omega & 0 \\ 0 & \omega^2 \end{pmatrix}, \quad C = \begin{pmatrix} \omega^2 & 0 \\ 0 & \omega \end{pmatrix}, \quad D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad E = \begin{pmatrix} 0 & \omega^2 \\ \omega & 0 \end{pmatrix}, \quad F = \begin{pmatrix} 0 & \omega \\ \omega^2 & 0 \end{pmatrix},$$ where \(\omega\) and \(\omega^2\) are the complex cube roots of 1. It is given that \(M\) is a group under matrix multiplication.

1. Write down the elements of a subgroup of order 2. [1]
2. Explain why there is no element \(X\) of the group, other than \(A\), which satisfies the equation \(X^2 = A\). [2]
3. By finding \(BE\) and \(EB\), verify the closure property for the pair of elements \(B\) and \(E\). [4]
4. Find the inverses of \(B\) and \(E\). [3]
5. Determine whether the group \(M\) is isomorphic to the group \(N\) which is defined as the set of numbers \(\{1, 2, 4, 8, 7, 5\}\) under multiplication modulo 9. Justify your answer clearly. [3]

A group \(G\), of order 8, is generated by the elements \(a\), \(b\), \(c\). \(G\) has the properties $$a^2 = b^2 = c^2 = e, \quad ab = ba, \quad bc = cb, \quad ca = ac,$$ where \(e\) is the identity.

1. Using these properties and basic group properties as necessary, prove that \(abc = cba\). [2]

The operation table for \(G\) is shown below.

1. List all the subgroups of order 2. [2]
2. List five subgroups of order 4. [3]
3. Determine whether all the subgroups of \(G\) which are of order 4 are isomorphic. [2]

The group \(G\) has binary operation \(*\) and order \(p\), where \(p\) is a prime number.

1. Determine the number of distinct subgroups of \(G\) Fully justify your answer. [2 marks]
2. \(G\) contains an element \(g\) which has period \(p\)
   1. State the general name given to elements such as \(g\) [1 mark]
   2. State the name of a group that is isomorphic to \(G\) [1 mark]
3. \(G\) contains an element \(g^r\), where \(r < p\) Find, in terms of \(g\), \(r\) and \(p\), the inverse of \(g^r\) [2 marks]
4. In the case when \(p = 5\) and the binary operation \(*\) represents addition modulo 5, \(G\) contains the elements 0, 1, 2, 3 and 4
   1. Explain why \(G\) is closed. [1 mark]
   2. Complete the Cayley table for \((G, *)\) [1 mark]

      \(*\)

Which one of the following sets forms a group under the given binary operation? Tick \((\checkmark)\) one box. [1 mark]

Set	Binary Operation
\(\{1, 2, 3\}\)	Addition modulo 4	\(\square\)
\(\{1, 2, 3\}\)	Multiplication modulo 4	\(\square\)
\(\{0, 1, 2, 3\}\)	Addition modulo 4	\(\square\)
\(\{0, 1, 2, 3\}\)	Multiplication modulo 4	\(\square\)

\(T\) is the set \(\{1, 2, 3, 4\}\). A binary operation \(\bullet\) is defined on \(T\) such that \(a \bullet a = 2\) for all \(a \in T\). It is given that \((T, \bullet)\) is a group.

1. Deduce the identity element in \(T\), giving a reason for your answer. [2]
2. Find the value of \(1 \bullet 3\), showing how the result is obtained. [3]
3. 1. Complete a group table for \((T, \bullet)\). [2]
   2. State with a reason whether the group is abelian. [1]