• Order of an Element
The order of an element g in a group G is the smallest
positive integer n such that gn = 5 e. (In additive
notation, this would be ng = 0.)
If no such integer exists, we say that g has infinite order.
The order of an element g is denoted by |g|.
So, to find the order of a group element g, you need only
compute the sequence of products g, g2, g3, . . ., until
you reach the identity for the first time. The exponent of
this product (or coefficient if the operation is addition) is
the order of g. If the identity never appears in the
sequence, then g has infinite order.
EXAMPLE 1 Consider U (15) 5 {1, 2, 4, 7, 8, 11, 13, 14}
under multiplication modulo 15. This group has order 8.
To find the order of the element 7, say, we compute the
sequence 71 = 7, 72 = 4, 73 = 13, 74 = 1, so |7| = 4. To find
the order of 11,
we compute 111 = 11, 112 = 1, so |11| = 2. Similar
computations show that |1| = 1, |2| = 4, |4| = 2, |8| = 4,
|13| = 4, |14| = 2. [Here is a trick that makes these
calculations easier. Rather than compute the sequence
131, 132, 133, 134, we may observe that 13 = -2 mod 15,
so that 132 = (-2)2 = 4, 133 = -2. 4 = -8, 134 = (-2) (-8) = 1.]
†
• To Find order of an element under
Multiplication Modulo: -
Consider U (15) = {1,2,4,7,8,11,13,14} under
multiplication modulo 15.
To find order of 7.
71 =7; 72 =49 = 4; 73 = 343 = 13; 74 =2401 =1.
Hence order of 11 is 2 is|11| =2.
• To Find order of an element under
Addition Modulo: -
Consider Z10 = {0,1,2,3,4,5,6,7,8,9] under
addition modulo 10.
Now 21=2; 2+2=4; 2+2+2=6; 2+2+2+2=8;
2+2+2+2+2=10=0
Hence order of |2| = 5
• The order of an element in abstract
algebra has various applications:
1. Group Structure: It helps determine the
structure of a group by understanding the
behavior of its elements under repeated
operations.
2. Cyclic Groups: In cyclic groups, the order of
an element \( g \) is the same as the number of
elements in the subgroup generated by \( g \).
This is fundamental in studying cyclic groups.
3. Subgroup Classification: The order of an
element helps classify subgroups of a group,
especially in determining if a subgroup is cyclic.
4. Cryptographic Applications: In cryptography,
the order of elements in certain groups is used
in algorithms for generating keys and
performing secure computations.
5. Permutation Groups: In permutation groups,
the order of an element corresponds to the
length of its cycle.
Understanding the order of elements in a group
is essential for solving problems in abstract
algebra and its applications in various fields.
