Prove group-theoretic identities

3 A multiplicative group contains the distinct elements \(e , x\) and \(y\), where \(e\) is the identity.

1. Prove that \(x ^ { - 1 } y ^ { - 1 } = ( y x ) ^ { - 1 }\).
2. Given that \(x ^ { n } y ^ { n } = ( x y ) ^ { n }\) for some integer \(n \geqslant 2\), prove that \(x ^ { n - 1 } y ^ { n - 1 } = ( y x ) ^ { n - 1 }\).
3. If \(x ^ { n - 1 } y ^ { n - 1 } = ( y x ) ^ { n - 1 }\), does it follow that \(x ^ { n } y ^ { n } = ( x y ) ^ { n }\) ? Give a reason for your answer.

3 A non-commutative group \(G\) consists of the six elements \(\left\{ e , a , a ^ { 2 } , b , a b , b a \right\}\) where \(e\) is the identity element, \(a\) is an element of order 3 and \(b\) is an element of order 2 .
By considering the row in \(G\) 's group table in which each of the above elements is pre-multiplied by \(b\), show that \(b a ^ { 2 } = a b\).

8

1. Explain why all groups of even order must contain at least one self-inverse element (that is, an element of order 2).
2. Prove that any group, in which every (non-identity) element is self-inverse, is abelian.
3. A student believes that, if \(x\) and \(y\) are two distinct, non-identity, self-inverse elements of a group, then the element \(x y\) is also self-inverse. The table shown here is the Cayley table for the non-cyclic group of order 6, having elements \(i , a , b , c , d\) and \(e\), where \(i\) is the identity.

   	\(i\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
   \(i\)	\(i\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
   \(a\)	\(a\)	\(i\)	\(d\)	\(e\)	\(b\)	\(c\)
   \(b\)	\(b\)	\(e\)	\(i\)	\(d\)	\(c\)	\(a\)
   \(c\)	\(c\)	\(d\)	\(e\)	\(i\)	\(a\)	\(b\)
   \(d\)	\(d\)	\(c\)	\(a\)	\(b\)	\(e\)	\(i\)
   \(e\)	\(e\)	\(b\)	\(c\)	\(a\)	\(i\)	\(d\)

   By considering the elements of this group, produce a counter-example which proves that this student is wrong.
4. A group \(G\) has order \(4 n + 2\), for some positive integer \(n\), and \(i\) is the identity element of \(G\). Let \(x\) and \(y\) be two distinct, non-identity, self-inverse elements of \(G\). By considering the set \(\mathrm { H } = \{ \mathrm { i } , \mathrm { x } , \mathrm { y } , \mathrm { xy } \}\), prove by contradiction that not all elements of \(G\) are self-inverse.

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 \(H\).

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.

3 The function min \(( a , b )\) is defined by: $$\begin{aligned} \min ( a , b ) & = a , a < b \\ & = b , \text { otherwise } \end{aligned}$$ For example, \(\min ( 7,2 ) = 2\) and \(\min ( - 4,6 ) = - 4\). Gary claims that the binary operation \(\Delta\), which is defined as $$x \Delta y = \min ( x , y - 3 )$$ where \(x\) and \(y\) are real numbers, is associative as finding the smallest number is not affected by the order of operation. Disprove Gary's claim.
[0pt] [2 marks]

6 A group \(G\) has order 12.

1. State, with a reason, the possible orders of the elements of \(G\). The identity element of \(G\) is \(e\), and \(x\) and \(y\) are distinct, non-identity elements of \(G\) satisfying the three conditions
   (1) \(\quad x\) has order 6 ,
   (2) \(x ^ { 3 } = y ^ { 2 }\),
   (3) \(x y x = y\).
2. Prove that \(y x ^ { 2 } y = x\).
3. Prove that \(G\) is not a cyclic group.

8 Let \(G = \left\{ g _ { 1 } , g _ { 2 } , g _ { 3 } , \ldots , g _ { n } \right\}\) be a finite abelian group of order \(n\) under a multiplicative binary operation, where \(g _ { 1 } = e\) is the identity of \(G\).

1. Let \(x \in G\). Justify the following statements:
   1. \(x g _ { i } = x g _ { j } \Leftrightarrow g _ { i } = g _ { j }\);
   2. \(\left\{ x g _ { 1 } , x g _ { 2 } , x g _ { 3 } , \ldots , x g _ { n } \right\} = G\).
   3. By considering the product of all \(G\) 's elements, and using the result of part (i)(b), prove that \(x ^ { n } = e\) for each \(x \in G\).
   4. Explain why
      (a) this does not imply that all elements of \(G\) have order \(n\),
      (b) this argument cannot be used to justify the same result for non-abelian groups.

9

1. Explain why all groups of even order must contain at least one self-inverse element (that is, an element of order 2).
2. Prove that any group in which every non-identity element is self-inverse is abelian.
3. Simon believes that if \(x\) and \(y\) are two distinct self-inverse elements of a group, then the element \(x y\) is also self-inverse. By considering the group of the six permutations of \(\left( \begin{array} { l l } 1 & 2 \end{array} \right)\), produce a counter-example to prove him wrong.
4. A group \(G\) has order \(4 n + 2\), for some positive integer \(n\), and \(i\) is the identity element of \(G\). Let \(x\) and \(y\) be two distinct self-inverse elements of \(G\). By considering the set \(H = \{ i , x , y , x y \}\), prove by contradiction that \(G\) cannot contain all self-inverse elements.

6 A group \(G\) has order 12.

1. State, with a reason, the possible orders of the elements of \(G\). The identity element of \(G\) is \(e\), and \(x\) and \(y\) are distinct, non-identity elements of \(G\) satisfying the three conditions
   (1) \(x\) has order 6 ,
   (2) \(x ^ { 3 } = y ^ { 2 }\),
   (3) \(x y x = y\).
2. Prove that \(y x ^ { 2 } y = x\).
3. Prove that \(G\) is not a cyclic group.

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]

The binary operation \(*\) is defined as $$a * b = ab + 1 \quad \text{where } a, b \in \mathbb{R}$$

1. Prove that \(*\) is commutative on \(\mathbb{R}\) [2 marks]
2. Prove that \(*\) is not associative on \(\mathbb{R}\) [3 marks]

The binary operation \(\nabla\) is defined as \(a \nabla b = a + b + ab\) where \(a, b \in \mathbb{R}\)

1. Determine if \(\nabla\) is commutative on \(\mathbb{R}\) Fully justify your answer. [2 marks]
2. Prove that \(\nabla\) is associative on \(\mathbb{R}\) [3 marks]

The binary operation \(\oplus\) acts on the positive integers \(x\) and \(y\) such that $$x \oplus y = x + y + 8 \pmod{k^2 - 16k + 74}$$ where \(k\) is a positive integer.

1. 1. Show that \(\oplus\) is commutative. [1 mark]
   2. Determine whether or not \(\oplus\) is associative. Fully justify your answer. [2 marks]
2. Find the values of \(k\) for which 3 is an identity element for the set of positive integers under \(\oplus\) [3 marks]