8.03f Subgroups: definition and tests for proper subgroups

8 The set \(X\) consists of all \(2 \times 2\) matrices of the form \(\left( \begin{array} { r r } x & - y \\ y & x \end{array} \right)\), where \(x\) and \(y\) are real numbers which are not both zero.

1. (a) The matrices \(\left( \begin{array} { c c } a & - b \\ b & a \end{array} \right)\) and \(\left( \begin{array} { c c } c & - d \\ d & c \end{array} \right)\) are both elements of \(X\). Show that \(\left( \begin{array} { c c } a & - b \\ b & a \end{array} \right) \left( \begin{array} { c c } c & - d \\ d & c \end{array} \right) = \left( \begin{array} { c c } p & - q \\ q & p \end{array} \right)\) for some real numbers \(p\) and \(q\) to be found in terms of \(a , b , c\) and \(d\).
   (b) Prove by contradiction that \(p\) and \(q\) are not both zero.
2. Prove that \(X\), under matrix multiplication, forms a group \(G\). [You may use the result that matrix multiplication is associative.]
3. Determine a subgroup of \(G\) of order 17.

4 A binary operation, ○, is defined on a set of numbers, \(A\), in the following way. \(a \circ b = \mathrm { k } _ { 1 } \mathrm { a } - \mathrm { k } _ { 2 } \mathrm {~b} + \mathrm { k } _ { 3 }\), for \(a , b \in A\),
where \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\) are constants (which are not necessarily in \(A\) ) and the operations addition, subtraction and multiplication of numbers have their usual notation and meaning. You are initially given the following information about ○ and \(A\).

• \(A = \mathbb { R }\)
• \(0 \circ 0 = 2\)
• An identity element, \(e\), exists for ∘ in \(A\)

4 The set \(G\) is given by \(G = \{ \mathbf { M } : \mathbf { M }\) is a real \(2 \times 2\) matrix and det \(\mathbf { M } = 1 \}\).

1. Show that \(G\) forms a group under matrix multiplication, × . You may assume that matrix multiplication is associative.
2. The matrix \(\mathbf { A } _ { n }\) is defined by \(\mathbf { A } _ { n } = \left( \begin{array} { l l } 1 & 0 \\ n & 1 \end{array} \right)\) for any integer \(n\). The set \(S\) is defined by \(\mathrm { S } = \left\{ \mathrm { A } _ { \mathrm { n } } : \mathrm { n } \in \mathbb { Z } , \mathrm { n } \geqslant 0 \right\}\).
   1. Determine whether \(S\) is closed under × .
   2. Determine whether \(S\) is a subgroup of ( \(G , \times\) ).
3. 1. Find a subgroup of ( \(G , \times\) ) of order 2 .
   2. By considering the inverse of the non-identity element in any such subgroup, or otherwise, show that this is the only subgroup of ( \(G , \times\) ) of order 2. The set of all real \(2 \times 2\) matrices is denoted by \(H\).
4. With the help of an example, explain why ( \(H , \times\) ) is not a group.

4

1. In each of the following cases, a set \(G\) and a binary operation ∘ are given. The operation ∘ may be assumed to be associative on \(G\). Determine which, if any, of the other three group axioms are satisfied by ( \(G , \circ\) ) and which, if any, are not satisfied.
   1. \(G = \{ 2 n + 1 : n \in \mathbb { Z } \}\) and \(\circ\) is addition.
   2. \(G = \{ a + b \sqrt { 2 } : a , b \in \mathbb { Z } \}\) and ∘ is multiplication.
   3. \(G\) is the set of all real numbers and ∘ is multiplication.
2. A group \(M\) consists of eight \(2 \times 2\) matrices under the operation of matrix multiplication. Five of the eight elements of \(M\) are as follows. $$\frac { 1 } { \sqrt { 2 } } \left( \begin{array} { l l } 1 & \mathrm { i } \\ \mathrm { i } & 1 \end{array} \right) \quad \frac { 1 } { \sqrt { 2 } } \left( \begin{array} { r r } - 1 & \mathrm { i } \\ \mathrm { i } & - 1 \end{array} \right) \quad \frac { 1 } { \sqrt { 2 } } \left( \begin{array} { r r } 1 & - \mathrm { i } \\ - \mathrm { i } & 1 \end{array} \right) \quad \left( \begin{array} { l l } 0 & \mathrm { i } \\ \mathrm { i } & 0 \end{array} \right) \quad \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$$
   1. Find the other three elements of \(M\). \(( N , * )\) is another group of order 8, with identity element \(e\). You are given that \(N = \langle a , b , c \rangle\) where \(a * a = b * b = c * c = e\).
   2. State whether \(M\) and \(N\) are isomorphic to each other, giving a reason for your answer.

1. 1. The table below is a Cayley table for the group \(G\) with operation ∘

1. State which element is the identity of the group.
2. Determine the inverse of the element ( \(b \circ c\) )
3. Give a reason why the set \(\{ a , b , e , f \}\) cannot be a subgroup of \(G\). You must justify your answer.
4. Show that the set \(\{ b , d , f \}\) is a subgroup of \(G\).
   (ii) Given that \(H\) is a group with an element \(x\) of order 3 and an element \(y\) of order 6 satisfying $$y x = x y ^ { 5 }$$ show that \(y ^ { 3 } x y ^ { 3 } x ^ { 2 }\) is the identity element. \includegraphics[max width=\textwidth, alt={}, center]{7d269bf1-f481-46bd-b9d3-fea211b186cf-02_2270_54_309_1980}

1. 1. A binary operation * is defined on positive real numbers by

$$a * b = a + b + a b$$ Prove that the operation * is associative.
(ii) The set \(G = \{ 1,2,3,4,5,6 \}\) forms a group under the operation of multiplication modulo 7

1. Show that \(G\) is cyclic. The set \(H = \{ 1,5,7,11,13,17 \}\) forms a group under the operation of multiplication modulo 18
2. List all the subgroups of \(H\).
3. Describe an isomorphism between \(G\) and \(H\).

6. \begin{figure}[h] \end{figure} Figure 3 shows a plane shape made up of a regular hexagon with an equilateral triangle joined to each edge and with alternate equilateral triangles shaded. The symmetries of this shape are the rotations and reflections of the plane that preserve the shape and its shading. The symmetries of the shape can be represented by permutations of the six vertices labelled 1 to 6 in Figure 3. The set of these permutations with the operation of composition form a group, \(G\).

1. Describe geometrically the symmetry of the shape represented by the permutation $$\left( \begin{array} { l l l l l l } 1 & 2 & 3 & 4 & 5 & 6 \\ 3 & 4 & 5 & 6 & 1 & 2 \end{array} \right)$$
2. Write down, in similar two-line notation, the remaining elements of the group \(G\).
3. Explain why each of the following statements is false, making your reasoning clear.
   1. \(G\) has a subgroup of order 4
   2. \(G\) is cyclic. Diagram 1, on page 23, shows an unshaded shape with the same outline as the shape in Figure 3.
4. Shade the shape in Diagram 1 in such a way that the group of symmetries of the resulting shaded shape is isomorphic to the cyclic group of order 6
   \includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_426_378_1464_845}
   \section*{Diagram 1} \section*{Spare copy of Diagram 1}
   \includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_424_375_2119_845}
   Only use this diagram if you need to redraw your answer to part (d).

1. 1. A group \(G\) contains distinct elements \(a , b\) and \(e\) where \(e\) is the identity element and the group operation is multiplication.

Given \(a ^ { 2 } b = b a\), prove \(a b \neq b a\) (ii) The set \(H = \{ 1,2,4,7,8,11,13,14 \}\) forms a group under the operation of multiplication modulo 15

1. Find the order of each element of \(H\).
2. Find three subgroups of \(H\) each of order 4, and describe each of these subgroups. The elements of another group \(J\) are the matrices \(\left( \begin{array} { c c } \cos \left( \frac { k \pi } { 4 } \right) & \sin \left( \frac { k \pi } { 4 } \right) \\ - \sin \left( \frac { k \pi } { 4 } \right) & \cos \left( \frac { k \pi } { 4 } \right) \end{array} \right)\) where \(k = 1,2,3,4,5,6,7,8\) and the group operation is matrix multiplication.
3. Determine whether \(H\) and \(J\) are isomorphic, giving a reason for your answer.

4

1. The binary operation is defined on \(\mathbb { Z }\) by \(a\) b \(b = a + b - a b\) for all \(a , b \in \mathbb { Z }\). Prove that is associative on \(\mathbb { Z }\). The operation ∘ is defined on the set \(A = \{ 0,2,3,4,5,6 \}\) by \(a \circ b = a + b - a b ( \bmod 7 )\) for all \(a , b \in A\).
2. Complete the Cayley table for \(\left( A , { } ^ { \circ } \right)\) given in the Printed Answer Booklet.
3. Prove that \(( A , \circ )\) is a group. You may assume that the operation is associative.
4. List all the subgroups of \(( A , \circ )\).

4 The group \(G\) consists of the symmetries of the equilateral triangle \(A B C\) under the operation of composition of transformations (which may be assumed to be associative). Three elements of \(G\) are

• \(\boldsymbol { i }\), the identity
• \(\boldsymbol { j }\), the reflection in the vertical line of symmetry of the triangle
• \(\boldsymbol { k }\), the anticlockwise rotation of \(120 ^ { \circ }\) about the centre of the triangle.

These are shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_204_531_735_772} \includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_211_543_975_762} \includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_216_543_1215_762}

1. Explain why the order of \(G\) is 6 .
2. Determine
   • the order of \(\boldsymbol { j }\),
   • the order of \(\boldsymbol { k }\).
   • - Express, in terms of \(\boldsymbol { j }\) and/or \(\boldsymbol { k }\), each of the remaining three elements of \(G\).
   • Draw a diagram for each of these elements.
   • Is the operation of composition of transformations on \(G\) commutative? Justify your answer.
   • List all the proper subgroups of \(G\).

5 The set \(S\) is defined as $$S = \{ A , B , C , D \}$$ where \(A = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right] \quad B = \left[ \begin{array} { c c } 0 & - 1 \\ 1 & 0 \end{array} \right] \quad C = \left[ \begin{array} { c c } - 1 & 0 \\ 0 & - 1 \end{array} \right] \quad D = \left[ \begin{array} { c c } 0 & 1 \\ - 1 & 0 \end{array} \right]\) The group \(G\) is formed by \(S\) under matrix multiplication.
The group \(H\) is defined as \(H = ( \langle \mathrm { i } \rangle , \times )\), where \(\mathrm { i } ^ { 2 } = - 1\) 5

1. 1. Prove that \(B\) is a generator of \(G\).
      Fully justify your answer.
      5
      1. (ii) Show that \(G \cong H\).
         Fully justify your answer.
         5
   2. 1. Explain why \(H\) has no subgroups of order 3
         Fully justify your answer.
         5
   3. (ii) Find all of the subgroups of \(H\).

9 The group \(\left( C , + _ { 4 } \right)\) contains the elements \(0,1,2\) and 3 9

1. 1. Show that \(C\) is a cyclic group.
      9
      1. (ii) State the group of symmetries of a regular polygon that is isomorphic to \(C\) 9
   2. The group ( \(V , \otimes\) ) contains the elements (1, 1), (1, -1), (-1, 1) and (-1, -1) The binary operation \(\otimes\) between elements of \(V\) is defined by $$( a , b ) \otimes ( c , d ) = ( a \times c , b \times d )$$ 9
   3. 1. Find the element in \(V\) that is the inverse of \(( - 1,1 )\) Fully justify your answer.
         [0pt] [2 marks]
         9
   4. (ii) Determine, with a reason, whether or not \(C \cong V\) \(\mathbf { 9 }\) (c) The group \(G\) has order 16
      Rachel claims that as \(1,2,4,8\) and 16 are the only factors of 16 then, by Lagrange's theorem, the group \(G\) will have exactly 5 distinct subgroups, including the trivial subgroup and \(G\) itself. Comment on the validity of Rachel's claim. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-16_2493_1721_214_150}

5 The set \(S\) consists of all \(2 \times 2\) matrices having determinant 1 or - 1 . For instance, the matrices \(\mathbf { P } = \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \\ - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right) , \mathbf { Q } = - \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \\ - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right)\) and \(\mathbf { R } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\) are elements of \(S\). It is given that \(\times _ { \mathbf { M } }\) is the operation of matrix multiplication.

1. State the identity element of \(S\) under \(\times _ { \mathbf { M } }\). The group \(G\) is generated by \(\mathbf { P }\), under \(\times _ { \mathbf { M } }\).
2. Determine the order of \(G\). The group \(H\) is generated by \(\mathbf { Q }\) and \(\mathbf { R }\), also under \(\times _ { \mathbf { M } }\).
3. 1. By finding each element of \(H\), determine the order of \(H\).
   2. List all the proper subgroups of \(H\).
4. State whether each of the following statements is true or false. Give a reason for each of your answers.

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 \(G\).

7 The multiplicative group \(G\) has eight elements \(e , a , b , c , a b , a c , b c , a b c\), where \(e\) is the identity. The group is commutative, and the order of each of the elements \(a , b , c\) is 2 .

1. Find four subgroups of \(G\) of order 4.
2. Give a reason why no group of order 8 can have a subgroup of order 3 . The group \(H\) has elements \(0,1,2 , \ldots , 7\) with group operation addition modulo 8 .
3. Find the order of each element of \(H\).
4. Determine whether \(G\) and \(H\) are isomorphic and justify your conclusion.

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]

In this question \(G\) is a group of order \(n\), where \(3 \leqslant n < 8\).

1. In each case, write down the smallest possible value of \(n\):
   1. if \(G\) is cyclic, [1]
   2. if \(G\) has a proper subgroup of order 3, [1]
   3. if \(G\) has at least two elements of order 2. [1]
2. Another group has the same order as \(G\), but is not isomorphic to \(G\). Write down the possible value(s) of \(n\). [2]

\(Q\) is a multiplicative group of order 12.

1. Two elements of \(Q\) are \(a\) and \(r\). It is given that \(r\) has order 6 and that \(a^2 = r^3\). Find the orders of the elements \(a\), \(a^2\), \(a^3\) and \(r^2\). [4]

The table below shows the number of elements of \(Q\) with each possible order. \(G\) and \(H\) are the non-cyclic groups of order 4 and 6 respectively.

1. Construct two tables, similar to the one above, to show the number of elements with each possible order for the groups \(G\) and \(H\). Hence explain why there are no non-cyclic proper subgroups of \(Q\). [5]

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]