Question

For each of the following sets, determine whether they form a group under the operation indicated (where it is relevant you may assume that matrix multiplication is associative): (a) the integers (mod 10 ) under addition; (b) the integers (mod 10 ) under multiplication; (c) the integers \(1,2,3,4,5,6\) under multiplication (mod 7 ); (d) the integers \(1,2,3,4,5\) under multiplication (mod 6 ); (e) all matrices of the form $$ \left(\begin{array}{cc} a & a-b \\ 0 & b \end{array}\right) $$ where \(a\) and \(b\) are integers (mod 5 ), and \(a \neq 0 \neq b\), under matrix multiplication; (f) those elements of the set in (e) that are of order 1 or 2 (taken together); (g) all matrices of the form \(\left(\begin{array}{lll}1 & 0 & 0 \\ a & 1 & 0 \\ b & c & 1\end{array}\right) \quad\) where \(a, b, c\) are integers, under matrix multiolication

Short answer

(a) Yes; (b) No; (c) Yes; (d) No; (e) Yes; (f) Yes; (g) Yes.

Step-by-step solution

Understanding Groups

A set forms a group under a binary operation if it satisfies four properties: closure, associativity, identity element, and invertibility.

Set (a) - Integers (mod 10) under addition

Check closure: Addition of any two integers (mod 10) is also an integer (mod 10). Check associativity: Addition is associative. Check identity: 0 is the identity element. Check invertibility: Each integer (mod 10) has an additive inverse within the set. Hence, it forms a group.

Set (b) - Integers (mod 10) under multiplication

Check closure: Multiplication of any two integers (mod 10) is also an integer (mod 10). Check associativity: Multiplication is associative. Check identity: 1 is the identity element. Check invertibility: Not all elements have a multiplicative inverse within the set (e.g., 2 does not have an inverse). Hence, it does not form a group.

Set (c) - Integers 1, 2, 3, 4, 5, 6 under multiplication (mod 7)

Check closure: Multiplication of any two elements of the set is still in the set after taking mod 7. Check associativity: Multiplication is associative. Check identity: 1 is the identity element. Check invertibility: Each element has a multiplicative inverse under mod 7. Hence, it forms a group.

Set (d) - Integers 1, 2, 3, 4, 5 under multiplication (mod 6)

Check closure: Multiplication of integers mod 6 does not always remain in the set (e.g., 2*3 mod 6 = 0, which is not in the set). Hence, it does not form a group.

Set (e) - Certain 2x2 matrices under matrix multiplication

Check closure: Given matrices multiply to form another matrix of the same form. Check associativity: Matrix multiplication is associative. Check identity: The identity matrix is of the same form. Check invertibility: Each matrix has an inverse. Hence, it forms a group.

Set (f) - Elements of (e) with order 1 or 2

Elements of order 1 or 2 must be the identity or its inverse. Check closure: Product of two such elements is still a valid element. Check associativity: Inherited from matrix multiplication. Check identity: The identity matrix is considered. Check invertibility: Given matrices are either identity or their inverse. Hence, this forms a group.

Set (g) - Certain 3x3 matrices under matrix multiplication

Check closure: Given matrices multiply to form another matrix of the same form. Check associativity: Matrix multiplication is associative. Check identity: The identity matrix of the same form exists. Check invertibility: Given matrices have an inverse. Hence, it forms a group.

Key concepts

Binary Operation

In mathematics, a binary operation is a calculation that combines two elements (called operands) to produce another element. Typical examples are addition and multiplication. Formally, a binary operation on a set S is a function that takes any two elements of S and returns another element of S. The key aspects of binary operations include:

* **Definition:** A binary operation \(\bullet\) on a set S can be written as: \(\bullet : S \times S \rightarrow S\) , where \(\times\) denotes a Cartesian product. The outcome of applying the operation to elements a and b of S is written as \(a \bullet b\).
* **Examples:**
1. For integers under addition, \(a + b\) represents the sum of a and b, which is also an integer.
2. Matrix multiplication involves taking two matrices and producing another matrix as the result.

Understanding binary operations is the first step in analyzing whether a set and operation can form a group.

Closure

Closure is one of the fundamental properties a set must satisfy to be considered a group under a given operation. It ensures that applying the operation to any two elements of the set results in another element within the same set. This concept ensures the internal consistency of the set with respect to the operation.

Let's look at some examples:
* **Example (a) Integers (mod 10) under addition:** Adding any two integers modulo 10 results in another integer modulo 10. For instance, \(3 + 7 \bmod 10 = 0\), which is still within the set \(\text{0, 1, 2, ..., 9}\). Hence, closure holds.
* **Example (d) Integers 1, 2, 3, 4, 5 under multiplication (mod 6):** Multiplying two elements like \(2 \times 3 \bmod 6 = 0\), results in 0, which is not in the original set \(1, 2, 3, 4, 5\). Thus, it fails the closure property.

For a set to be closed under an operation, every combination of elements under that operation must produce an element still within the set.

Identity Element

An identity element is a special type of element within a set that, when combined with any element in the set using the binary operation, leaves that element unchanged. It is a vital component in the group structure since it provides a baseline for combining elements.

Specific points to note include:
* **Definition:** An element \(e\) in a set S is called an identity element under a binary operation \(\bullet\) if for every element \(a\) in S, the equation \(a \bullet e = e \bullet a = a\) holds.

* **Example (a) Integers (mod 10) under addition:** Here, the identity element is 0 because \((a + 0) \bmod 10 = a\) for any integer a.
* **Example (b) Integers (mod 10) under multiplication:** The identity element is 1 since \( (a \times 1) \bmod 10 = a\) for any integer a.

The identity element must always exist within the set for it to be considered for group formation.

Inverse Element

Every element in a group must have an inverse element. The inverse of an element under a given operation essentially 'undoes' the effect of the element, returning the identity element when the operation is applied.

Important points include:
* **Definition:** An element \(a'\) in a set S is called the inverse of element \(a\) under a binary operation \(\bullet\) if \(a \bullet a' = a' \bullet a = e\), where e is the identity element of the set under operation \(\bullet\).

* **Example (a) Integers (mod 10) under addition:** Each integer has an additive inverse within the set. For instance, the inverse of 7 is 3 because \((7 + 3) \bmod 10 = 0\), where 0 is the identity element.
* **Example (b) Integers (mod 10) under multiplication:** Not all integers have a multiplicative inverse. For instance, there's no integer \(x\) such that \((2 \times x) \bmod 10 = 1\). Thus, it doesn't satisfy the inverse property for all elements and doesn't form a group.

The presence of inverse elements ensures that there is a way to revert to the identity element, a critical aspect of group theory.