8.03f Subgroups: definition and tests for proper subgroups

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]

      \(*\)

\(G\) 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 \(X\) consists of all \(2 \times 2\) matrices of the form \(\begin{pmatrix} x & -y \\ y & x \end{pmatrix}\), where \(x\) and \(y\) are real numbers which are not both zero.

1. 1. The matrices \(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\) and \(\begin{pmatrix} c & -d \\ d & c \end{pmatrix}\) are both elements of \(X\). Show that \(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\begin{pmatrix} c & -d \\ d & c \end{pmatrix} = \begin{pmatrix} p & -q \\ q & p \end{pmatrix}\) for some real numbers \(p\) and \(q\) to be found in terms of \(a\), \(b\), \(c\) and \(d\). [2]
   2. Prove by contradiction that \(p\) and \(q\) are not both zero. [5]
2. Prove that \(X\), under matrix multiplication, forms a group \(G\). [You may use the result that matrix multiplication is associative.] [4]
3. Determine a subgroup of \(G\) of order 17. [2]

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

1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). [5]
2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3. [4]

The group \(G\), of order 8, consists of the elements \(\{e, a, b, c, ab, bc, ca, abc\}\), together with a multiplicative binary operation, where \(e\) is the identity and $$a^2 = b^2 = c^2 = e, \quad ab = ba, \quad bc = cb \quad \text{and} \quad ca = ac.$$

1. Construct the group table of \(G\). [You are not required to show how individual elements of the table are determined.] [4]
2. List all the proper subgroups of \(G\). [5]

A group \(G\) has distinct elements \(e, a, b, c, \ldots\), where \(e\) is the identity element and \(\circ\) is the binary operation. Prove that if $$a \circ a = b, \quad b \circ b = a$$ then the set of elements \(\{e, a, b\}\) forms a subgroup of \(G\). [5] Prove that if $$a \circ a = b, \quad b \circ b = c, \quad c \circ c = a$$ then the set of elements \(\{e, a, b, c\}\) does not form a subgroup of \(G\). [5]