Order Of an Element
Duration: 6 min
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The video lecture provides a comprehensive introduction to the concept of the "Order of an element" within the context of group theory. It begins by defining the order of an element 'a' in a group (A, *) as the smallest positive integer 'n' such that a^n equals the identity element 'e', denoted as O(a). The instructor outlines four key properties: the identity element always has an order of one; an element and its inverse share the same order; and in infinite groups, the order is infinite, except for the identity. The lecture then transitions to a practical application, posing a question to find the order of each element in the group {0, 1, 2, 3} under addition modulo 4 (+4). The instructor systematically constructs the Cayley table for this group and calculates the order for every element, demonstrating the theoretical concepts with concrete arithmetic.
Chapters
0:00 – 2:00 00:00-02:00
The session opens with a slide titled "Order of an element," defining the notation O(a) for an element 'a' in a group (A, *). The instructor explains the first property: O(a) = n, where 'n' is the smallest positive integer such that a^n = e. He lists three additional properties: the order of the identity element is always one; the order of an element and its inverse is always the same; and in an infinite group, the order does not exist or is infinite, except for the identity. Around the 1:50 mark, the slide changes to a specific problem: "Q consider a group {0,1,2,3}, +4 and find the order of each element?" A blank Cayley table for the operation +4 is displayed, setting the stage for the practical demonstration.
2:00 – 5:00 02:00-05:00
The instructor begins filling out the Cayley table for the group {0, 1, 2, 3} under addition modulo 4. He fills the first row and column with 0, 1, 2, 3 respectively. He then calculates the remaining entries: 1+1=2, 1+2=3, 1+3=0; 2+1=3, 2+2=0, 2+3=1; and 3+1=0, 3+2=1, 3+3=2. With the table complete, he starts calculating the order of element 0. He writes 0^1 = 0, noting that since 0 is the identity, its order is 1. Next, he calculates the order of element 1. He writes 1^1 = 1, 1^2 = 1+1=2, 1^3 = 1+2=3, and 1^4 = 1+3=0. Since 1^4 equals the identity 0, he concludes that the order of 1 is 4.
5:00 – 6:24 05:00-06:24
Continuing the calculation, the instructor determines the order of element 2. He writes 2^1 = 2 and 2^2 = 2+2=0. Since 2^2 equals the identity 0, he concludes O(2) = 2. He then moves to element 3, writing 3^1 = 3, 3^2 = 3+3=2, 3^3 = 3+2=1, and 3^4 = 3+1=0. Because 3^4 equals the identity 0, he determines O(3) = 4. The instructor summarizes the findings on the board: O(0)=1, O(1)=4, O(2)=2, and O(3)=4. He points out that elements 1 and 3 are generators of the group because their order equals the total number of elements in the group. The video concludes with the completed table and all calculated orders visible.
The lecture effectively bridges theoretical definitions with practical computation. By defining the order of an element as the smallest power yielding the identity, the instructor establishes a clear criterion for calculation. The example of the group Z4 under addition modulo 4 serves as a perfect illustrative case. The step-by-step construction of the Cayley table ensures that the arithmetic operations are transparent. The calculation of orders for 0, 1, 2, and 3 reinforces the properties discussed earlier, specifically showing that the identity has order 1, elements can have orders less than the group size (like 2), and elements can have orders equal to the group size (like 1 and 3). This progression from definition to property to application provides a complete understanding of how to find the order of elements in a finite group.