8.03l Isomorphism: determine using informal methods

6

1. The set \(S\) consists of all \(2 \times 2\) matrices of the form \(\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)\), where \(n \in \mathbb { Z }\).
   1. Show that \(S\), under the operation of matrix multiplication, forms a group \(G\). [You may assume that matrix multiplication is associative.]
   2. State, giving a reason, whether \(G\) is abelian.
   3. The group \(H\) is the set \(\mathbb { Z }\) together with the operation of addition. Explain why \(G\) is isomorphic to \(H\).
   4. The plane transformation \(T\) is given by the matrix \(\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)\), where \(n\) is a non-zero integer. Describe \(T\) fully.

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.

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]

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]

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]

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]

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]

\(G\) is a group of order 8.

1. Explain why there is no subgroup of \(G\) of order 6. [1]

You are now given that \(G\) is a cyclic group with the following features: • \(e\) is the identity element of \(G\), • \(g\) is a generator of \(G\), • \(H\) is the subgroup of \(G\) of order 4.

1. Write down the possible generators of \(H\). [2]

\(M\) is the group \((\{0, 1, 2, 3, 4, 5, 6, 7\}, +_8)\) where \(+_8\) denotes the binary operation of addition modulo 8. You are given that \(M\) is isomorphic to \(G\).

1. Specify all possible isomorphisms between \(M\) and \(G\). [4]

The set \(G = \{1, 4, 5, 6, 7, 9, 11, 16, 17\}\) is a group of order 9 under the binary operation of multiplication modulo 19.

1. Show that \(G\) is a cyclic group generated by the element 4. [3]
2. Find another generator for \(G\). Justify your answer. [2]
3. Specify two distinct isomorphisms from the group \(J = \{0, 1, 2, 3, 4, 5, 6, 7, 8\}\) under addition modulo 9 to \(G\). [5]

A class of students is set the task of finding a group of functions, under composition of functions, of order 6. Student P suggests that this can be achieved by finding a function \(f\) for which \(f^6(x) = x\) and using this as a generator for the group.

1. Explain why the suggestion by Student P might not work. [2]

Student Q observes that their class has already found a group of order 6 in a previous task; a group consisting of the powers of a particular, non-singular \(2 \times 2\) real matrix \(\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), under the operation of matrix multiplication.

1. Explain why such a group is only possible if \(\det(\mathbf{M}) = 1\) or \(-1\). [2]
2. Write down values of \(a\), \(b\), \(c\) and \(d\) that would give a suitable matrix \(\mathbf{M}\) for which \(\mathbf{M}^6 = \mathbf{I}\) and \(\det(\mathbf{M}) = 1\). [1]

Student Q believes that it is possible to construct a rational function \(f\) in the form \(f(x) = \frac{ax + b}{cx + d}\) so that the group of functions is isomorphic to the matrix group which is generated by the matrix \(\mathbf{M}\) of part (iii).

1. 1. Write down and simplify the function \(f\) that, according to Student Q, corresponds to \(\mathbf{M}\). [1]
   2. By calculating \(\mathbf{M}^2\), show that Student Q's suggestion does not work. [2]
   3. Find a different function \(f\) that will satisfy the requirements of the task. [4]

\(G\) is the set \(\{2, 4, 6, 8\}\), \(H\) is the set \(\{1, 5, 7, 11\}\) and \(\times_n\) denotes the operation of multiplication modulo \(n\).

1. Construct the multiplication tables for \((G, \times_{10})\) and \((H, \times_{12})\). [2]
2. By verifying the four group axioms, show that \(G\) and \(H\) are groups under their respective binary operations, and determine whether \(G\) and \(H\) are isomorphic. [6]

[You may assume that \(\times_n\) is associative.]