Science and Technology Indonesia e-ISSN:2580-4391 p-ISSN:2580-4405 Vol.
No.
July 2022 Research Paper Evaluate All The Order of Every Element in The Higher Even.
Odd, and Prime Order of Group for Composition Md.
Abdul Mannan1 *.
Halima Akter1 .
Md.
Amanat Ullah2 1 Department of Mathematics.
Uttara University.
Dhaka, 1230.
Bangladesh 2 Department of Computer Science & Engineering.
Uttara University.
Dhaka, 1230.
Bangladesh *Corresponding author: mannan.
iu31@gmail.
Abstract This paper aims to treat a study on the order of every element in the higher even, odd and prime order of group for composition.
fact, express order of a group and order of an element of a group in real numbers.
Here we discuss the higher order of groups in different types of order, which will give us practical knowledge to see the applications of the composition.
In order to find out the order of an element am OOG in which an = e= identity element, then find the least common multiple .
(LCM))= yuI ) of m and n.
The least common multiple of two numbers is the "smallest non-zero common number," which is a multiple of both the numbers.
So O.
m )= yuI /m.
Also, if G is a finite group, n is a positive integer, and aOOG then the order of the products na.
When G is a finite group, every element must have finite order, but the converse is false.
There are infinite groups where each element has finite order.
Finally, find out the order of every element of a group in different types of the higher even, odd and prime order of group for composition.
Keywords Order of Element of a Group.
Multiplication Composition.
LCM Received: 9 March 2022.
Accepted: 14 June 2022 https://doi.
org/10.
26554/sti.
INTRODUCTION
We propose to study the groups of the order of an element of a group, the order of the group, the integral powers of an element of a group, etc.
Then discuss the order of every element in the higher even, odd and prime order of the group for composition.
The group notation is o or O .
We will frequently omit the symbol for the group operation, but we will also often write the operation as A or when it represents multiplication or addition in a group and write 1 or 0 for the corresponding identity elements, respectively.
Its addition , multiplication y or (.
) is used as binary operation.
If the group operation is denoted as a multiplication, then an element aOOG is said to be order n if n is the least positive integer such that an = e or o.
On i.
, if an = e and ar Oe OA rOON s.
r<n.
The order of a is denoted by o.
an Oe for any nOON, then a is said to be of zero-order or infinite order (Beltryn et al.
, 2.
Let e is the identity element in (G, ).
An element aOOG is said to be order n if nOOZ such that na= e or o.
On.
, if na= e and arOe OA rOON s.
0<r<n.
The order of a is denoted by o.
If naOe for any nOON, then a is said to be of zeroorder or infinite order (Gow, 2.
Then the classification of finite simple groups .
ee e.
Gorenstein and Lyons, 1983.
Gorenstein et al.
, 1999.
Robinson, 2.
comes into play, and one has to be able to handle the three different families of simple groups with appropriate techniques.
Nonetheless, the classification problem for finite groups into two problems: .
identify the simple groups.
identify the ways these simple groups may be put together to form bigger groups.
Next, we discuss the extension of the associative property to products with any number of factors as we use the multiplication-related theorem of a group of different orders.
As a result, we find out the order of a group of the higher even, odd, and prime order groups for multiplication composition.
But here, we discuss the order of a group of higher odd, even, and prime order of groups as like as 69, 70, and 71.
Then we find out the order of every element of a group in different types of the higher even, odd and prime order of group for composition (Mannan et al.
Hall Jr, 1967.
Brauer, 1942.
McKay and Wales, 1971.
Trefethen, 2.
INTEGRAL POWERS OF AN ELEMENT OF A
GROUP
1 Multiplication Composition (Morety, 2021.
Bychle, 2.
Let (G, .
) be a group.
Let aOOG be an arbitrary element.
closure property, all the elements a, aa, a.
A A A etc.
belong to Since the composition in G is associative.
Hence a.
A A A to n factors is independent of the manner in which the factors are Science and Technology Indonesia, 7 .
333-343 Mannan et.
If n is a positive integer, then define an = a A A to n factors an OOG, by closure property If e is identity in G, then we define a0 = e.
If n is a negative integer, then by define aOen =.
n ) Oe1 , where .
n ) Oe1 is the inverse of an .
Consequently, .
n ) Oe1 OOG, since the inverse of every element of G belongs to G.
O aOen OOG.
According to the definition .
n ) Oe1 = .
A A A to factor.
Oe1 = .
Oe1 a Oe1 a Oe1 ) A A A to Oe1 n = .
) Oa Oen = .
n ) Oe1 = .
Oe1 ) n The following law of indices can be easily proved.
m ) n = a mn a a =a OA a OOG and OA m, n OO Z a OOG and OA m, n OO Z Thus we defined an all integral values of n, positive, negative, or zero.
2 Addition Composition (Kurdachenko et al.
, 2019a.
Kurdachenko et al.
, 2020.
Let (G, ) be a group.
Let aOOG be an arbitrary element.
By closure property, all the elements a, a a, a a a.
A A A etc.
belong to G.
Since the composition in G is associative.
Hence, a a aA A A to n factors is independent of the manner in which the factors are If n is a positive integer, then define na= a a aA A A to n factors.
If e is identity in G, then we define 0.
a= e.
If n is a negative integer, then by define (-.
a= -.
, where -.
is the inverse of na.
Consequently, -.
OOG.
O (-.
aOOG.
According to the definition Oe.
= Oe.
a a A A A to = (Oe.
(Oe.
(Oe.
A A A to = n(Oe.
O (Oe.
a = Oe.
= n(Oe.
The following law of indices can be easily proved.
= .
a = ma na
a OOG
a OOG
SIGNIFICANCE OF THE ORDER OF AN ELEMENT OF A GROUP
We begin this section with the following theorem-related significance of the order of an element of a group.
1 Theorem (Robinson, 2012.
Kurdachenko et al.
, 2020.
Show that the order of every element of a finite group is finite.
Proof: Let G be a finite group with multiplication composition.
Let aOOG be an arbitrary element.
Now we will prove that o.
is finite.
By closure property, all the elements a2 = a.
a, a3 = a.
A A A etc.
belong to G.
a, a2 , a3 , a4 , a5 , a6 , a7 .
A A A etc.
belong to G.
But all these elements are not distinct since G is finite.
Let e be the identity in G, then a0 = e.
Let us suppose that am =an where m>n.
Neam aOen = an aOen = a0 = e NeamOen = eNeap = e, where p= m-n>0, as m>n Also, m and n are finite, and hence p is a finite positive integer.
Now p is a positive integer s.
ap = e.
This proves that o.
Op= finite number o.
finite number Ne o.
is finite.
Remark: The order of any element of a finite group can never exceed the order of the group.
2 Theorem (McCann, 2018.
Moscatiello, 2.
Show that the order of any integral power of an element of a group G is less than or equal to a.
m ) Oo.
OA aOOG and mOON.
Proof: Let aOOG be an arbitrary element s.
= n, where n is a natural number.
Such that an = e = identify of .
Let am be any power of a and let o.
m )= p Now we will prove that o.
m ) Oo.
pOn We have, o.
= n Ne an = e Ne anm = em = e Ne.
m ) n = e Ne o.
m ) On Ne pOn Remark: This theorem can also be expressed in the following The order of any integral power of an element a of a group cannot exceed the order of a.
If aOOG i.
G is a group, then o.
m ) Oo.
OA aOOG and mOON.
If G is a group and aOOG, then the order of any power of an element a is almost equal to the order of a.
and OA m, n OO Z and OA m, n OO Z Thus we defined na all integral values of n, positive, negative, or zero.
A 2022 The Authors.
Page 334 of 343 Mannan et.
RESULT AND DISCUSSION
Here, we discuss the result of an order of every element for multiplication composition in the higher even, odd and prime order of the group.
As usual, we can use a composition-related theorem to evaluate the order of a group of different orders, such as order , etc.
, i.
, whose order is not so high (Not HigherOrder Group.
As a result, we use a multiplication-related theorem to evaluate the order of a group of the higher odd, even, and prime order groups for composition.
For that region, here we discuss the order of a group of the higher odd, even, and prime order of group as like as 69, 70, and 71.
1 The Higher Odd Order of a Group for Multiplication Composition (Cook et al.
, 2019.
Kurdachenko et al.
Kurdachenko et al.
, 2020.
Find the order of every element in the multiplication group G= .
, a2 , a3 , a4 .
A A A , a69 = .
Solution:
We know that o.
m )= myuI , where yuI = l.
m of m and n To determine o.
2 ) Here, o.
69 )= e,l.
m of 69 and 2 is 138 Oo.
2 )= 138 2 = 69 Ne o.
2 )= 69 To determine o.
3 ) The identity element of the given group is a69 = e Ne o.
= 69 O o.
= 69 Here, o.
69 )= e,l.
m of 69 and 3 is 69 Oo.
3 )= 69 3 = 23 Ne o.
)= To determine o.
4 ) Here, o.
69 )= e,l.
m of 69 and 4 is 276 Oo.
4 )= 276 4 = 69 Ne o.
)= 69 To determine o.
5 ) Here, o.
69 )= e,l.
m of 69 and 5 is 345 Oo.
5 )= 345 5 = 69 Ne o.
5 )= 69 To determine o.
6 ) Here, o.
69 )= e,l.
m of 69 and 6 is 138 Oo.
6 )= 138 6 = 23 Ne o.
6 )= 23 To determine o.
7 ) Here, o.
69 )= e,l.
m of 69 and 7 is 483 Oo.
7 )= 483 = 69 Ne o.
7 )= 69 To determine o.
8 ) Here, o.
69 )= e,l.
m of 69 and 8 is 552 Oo.
8 )= 552 8 = 69 Ne o.
8 )= 69 To determine o.
9 ) Here, o.
69 )= e,l.
m of 69 and 9 is 207 Oo.
9 )= 207 9 = 23 Ne o.
)= 23 A 2022 The Authors.
Science and Technology Indonesia, 7 .
333-343 To determine o.
10 ) Here, o.
69 )= e,l.
m of 69 and 10 is 690 Oo.
10 )= 690 10 = 69 Ne o.
10 )= 69 To determine o.
11 ) Here, o.
69 )= e,l.
m of 69 and 11 is 759 Oo.
11 )= 759 11 = 69 Ne o.
)= 69 To determine o.
12 ) Here, o.
69 )= e,l.
m of 69 and 12 is 276 Oo.
12 )= 276 12 = 23 Ne o.
)= 23 To determine o.
13 ) Here, o.
69 )= e,l.
m of 69 and 13 is 897 Oo.
13 )= 897 13 = 69 Ne o.
13 )= 69 To determine o.
14 ) Here, o.
69 )= e,l.
m of 69 and 14 is 966 Oo.
14 )= 966 14 = 69 Ne o.
14 )= 69 To determine o.
15 ) Here, o.
69 )= e,l.
m of 69 and 15 is 345 Oo.
15 )= 345 12 = 23 Ne o.
)= 23 To determine o.
16 ) Here, o.
69 )= e,l.
m of 69 and 16 is 1104 Oo.
16 )= 1104
14 = 69
Ne o.
)= 69 To determine o.
17 ) Here, o.
69 )= e,l.
m of 69 and 17 is 1173 Oo.
17 )= 1173
= 69
Ne o.
17 )= 69 To determine o.
18 ) Here, o.
69 )= e,l.
m of 69 and 18 is 414 Oo.
18 )= 414 18 = 23 Ne o.
18 )= 23 To determine o.
19 ) Here, o.
69 )= e,l.
m of 69 and 19 is 1311 Oo.
19 )= 1311
19 = 69
Ne o.
19 )= 69 To determine o.
20 ) Here, o.
69 )= e,l.
m of 69 and 20 is 1380 Oo.
20 )= 1280
20 = 69
Ne o.
)= 69 To determine o.
21 ) Here, o.
69 )= e,l.
m of 69 and 21 is 483 Oo.
21 )= 483 21 = 23 Ne o.
)= 23 To determine o.
22 ) Here, o.
69 )= e,l.
m of 69 and 22 is 1518 Oo.
22 )= 1518
22 = 69
Ne o.
22 )= 69 To determine o.
23 ) Here, o.
69 )= e,l.
m of 69 and 23 is 1587 Oo.
23 )= 1587
23 = 69
Ne o.
23 )= 69 Page 335 of 343 Mannan et.
Science and Technology Indonesia, 7 .
333-343 To determine o.
24 ) Here, o.
69 )= e,l.
m of 69 and 24 is 552 Oo.
24 )= 552 24 = 23 Ne o.
24 )= 23 To determine o.
38 ) Here, o.
69 )= e,l.
m of 69 and 38 is 2622 Oo.
38 )= 2622
38 = 69
Ne o.
38 )= 69 To determine o.
25 ) Here, o.
69 )= e,l.
m of 69 and 25 is 1725 Oo.
25 )= 1725
25 = 69
Ne o.
)= 69 To determine o.
39 ) Here, o.
69 )= e,l.
m of 69 and 39 is 897 Oo.
39 )= 897 39 = 23 Ne o.
)= 23 To determine o.
26 ) Here, o.
69 )= e,l.
m of 70 and 26 is 910 Oo.
26 )= 910 26 = 35 Ne o.
)= 35 To determine o.
40 ) Here, o.
69 )= e,l.
m of 69 and 40 is 2760 Oo.
40 )= 2760
40 = 69
Ne o.
)= 69 To determine o.
27 ) Here, o.
69 )= e,l.
m of 69 and 27 is 621 Oo.
27 )= 621 = 23 Ne o.
27 )= 23 To determine o.
41 ) Here, o.
69 )= e,l.
m of 69 and 41 is 2829 Oo.
41 )= 2829
41 = 69
Ne o.
41 )= 69 To determine o.
28 ) Here, o.
69 )= e,l.
m of 69 and 28 is 1932 Oo.
28 )= 1932
28 = 69
Ne o.
28 )= 69 To determine o.
42 ) Here, o.
69 )= e,l.
m of 69 and 42 is 966 Oo.
42 )= 966 42 = 23 Ne o.
42 )= 23 To determine o.
29 ) Here, o.
69 )= e,l.
m of 69 and 29 is 2001 Oo.
29 )= 2001
29 = 69
Ne o.
29 )= 69 To determine o.
43 ) Here, o.
69 )= e,l.
m of 69 and 43 is 2967 Oo.
43 )= 2967
43 = 69
Ne o.
43 )= 69 To determine o.
30 ) Here, o.
69 )= e,l.
m of 69 and 30 is 690 Oo.
30 )= 690 30 = 23 Ne o.
)= 23 To determine o.
44 ) Here, o.
69 )= e,l.
m of 69 and 44 is 3036 Oo.
44 )= 3036
44 = 69
Ne o.
)= 69 To determine o.
31 ) Here, o.
69 )= e,l.
m of 69 and 31 is 2139 Oo.
31 )= 2139
31 = 69
Ne o.
)= 69 To determine o.
45 ) Here, o.
69 )= e,l.
m of 69 and 45 is 1035 Oo.
32 )= 1035
45 = 23
Ne o.
45 )= 23 To determine o.
32 ) Here, o.
69 )= e,l.
m of 69 and 32 is 2208 Oo.
32 )= 2208
32 = 69
Ne o.
32 )= 69 To determine o.
46 ) Here, o.
69 )= e,l.
m of 69 and 46 is 3174 Oo.
46 )= 3174
46 = 69
Ne o.
46 )= 69 To determine o.
33 ) Here, o.
69 )= e,l.
m of 69 and 33 is 759 Oo.
33 )= 759 33 = 23 Ne o.
32 )= 23 To determine o.
47 ) Here, o.
69 )= e,l.
m of 69 and 32 is 3243 Oo.
47 )= 3243
= 69
Ne o.
47 )= 69 To determine o.
34 ) Here, o.
69 )= e,l.
m of 69 and 34 is 2346 Oo.
34 )= 2346
34 = 69
Ne o.
34 )= 69 To determine o.
48 ) Here, o.
69 )= e,l.
m of 69 and 48 is 1104 Oo.
48 )= 1104
48 = 69
Ne o.
)= 69 To determine o.
35 ) Here, o.
69 )= e,l.
m of 69 and 35 is 2415 Oo.
35 )= 2415
35 = 69
Ne o.
)= 69 To determine o.
49 ) Here, o.
69 )= e,l.
m of 69 and 49 is 3381 Oo.
49 )= 2281
49 = 69
Ne o.
)= 69 To determine o.
36 ) Here, o.
69 )= e,l.
m of 69 and 36 is 828 Oo.
36 )= 828 36 = 23 Ne o.
36 )= 23 To determine o.
50 ) Here, o.
69 )= e,l.
m of 69 and 50 is 3450 Oo.
50 )= 3450
50 = 69
Ne o.
50 )= 69 To determine o.
37 ) Here, o.
69 )= e,l.
m of 69 and 37 is 2553 Oo.
37 )= 2553
= 69
Ne o.
37 )= 69 To determine o.
25 ) Here, o.
69 )= e,l.
m of 69 and 51 is 1173 Oo.
51 )= 1173
51 = 23
Ne o.
51 )= 23 A 2022 The Authors.
Page 336 of 343 Mannan et.
Science and Technology Indonesia, 7 .
333-343 To determine o.
52 ) Here, o.
69 )= e,l.
m of 69 and 52 is 3588 Oo.
52 )= 3588
52 = 69
Ne o.
52 )= 69 To determine o.
66 ) Here, o.
69 )= e,l.
m of 69 and 66 is 1518 Oo.
66 )= 1518
66 = 23
Ne o.
66 )= 23 To determine o.
53 ) Here, o.
69 )= e,l.
m of 69 and 53 is 3657 Oo.
53 )= 3657
53 = 69
Ne o.
)= 69 To determine o.
67 ) Here, o.
69 )= e,l.
m of 69 and 67 is 4623 Oo.
67 )= 4623
= 69
Ne o.
)= 69 To determine o.
54 ) Here, o.
69 )= e,l.
m of 69 and 54 is 1242 Oo.
54 )= 1242
54 = 23
Ne o.
)= 23 To determine o.
68 ) Here, o.
69 )= e,l.
m of 69 and 68 is 4692 Oo.
68 )= 4692
68 = 69
Ne o.
68 )= 69 To determine o.
55 ) Here, o.
69 )= e,l.
m of 69 and 55 is 3795 Oo.
55 )= 3795
55 = 69
Ne o.
55 )= 69 To determine o.
69 ) Here, o.
69 )= e,l.
m of 69 and 69 is 69 Oo.
69 )= 69 69 = 1 Ne o.
69 )= 1 To determine o.
56 ) Here, o.
69 )= e,l.
m of 69 and 56 is 3864 Oo.
56 )= 3864
56 = 69
Ne o.
56 )= 69 2 The Higher Even Order of a Group for Multiplication Composition (Kurdachenko et al.
, 2020c.
Russo, 2019.
Subbotin, 2.
Find the order of every element in the multiplication group G= .
, a2 , a3 , a4 .
A A A , a70 = .
To determine o.
57 ) Here, o.
69 )= e,l.
m of 69 and 57 is 1311 Oo.
57 )= 1311
= 23
Ne o.
57 )= 23 To determine o.
58 ) Here, o.
69 )= e,l.
m of 69 and 58 is 4002 Oo.
58 )= 4002
58 = 69
Ne o.
)= 69 To determine o.
59 ) Here, o.
69 )= e,l.
m of 69 and 59 is 4071 Oo.
59 )= 4071
59 = 69
Ne o.
)= 69 To determine o.
60 ) Here, o.
69 )= e,l.
m of 69 and 60 is 1380 Oo.
60 )= 1380
60 = 23
Ne o.
60 )= 23 To determine o.
61 ) Here, o.
69 )= e,l.
m of 69 and 61 is 4209 Oo.
61 )= 4209
61 = 69
Ne o.
61 )= 69 To determine o.
62 ) Here, o.
69 )= e,l.
m of 69 and 62 is 4278 Oo.
62 )= 4278
62 = 69
Ne o.
62 )= 69 To determine o.
63 ) Here, o.
69 )= e,l.
m of 69 and 63 is 1449 Oo.
63 )= 1449
63 = 23
Ne o.
)= 23 To determine o.
64 ) Here, o.
69 )= e,l.
m of 69 and 64 is 4416 Oo.
64 )= 4416
64 = 69
Ne o.
64 )= 69 To determine o.
65 ) Here, o.
69 )= e,l.
m of 69 and 65 is 4485 Oo.
65 )= 4485
65 = 69
Ne o.
65 )= 69 A 2022 The Authors.
Solution:
The identity element of the given group is a70 = e Ne o.
= 70 O o.
= 70 We know that o.
m )= myuI , where yuI = l.
m of m and n To determine o.
2 ) Here, o.
70 )= e,l.
m of 70 and 2 is 70 Oo.
2 )= 70 2 = 35 Ne o.
)= To determine o.
3 ) Here, o.
70 )= e,l.
m of 70 and 3 is 210 Oo.
3 )= 210 3 = 70 Ne o.
3 )= 70 To determine o.
4 ) Here, o.
70 )= e,l.
m of 70 and 4 is 140 Oo.
4 )= 140 4 = 35 Ne o.
4 )= 35 To determine o.
5 ) Here, o.
70 )= e,l.
m of 70 and 5 is 70 Oo.
5 )= 70 5 = 14 Ne o.
)= To determine o.
6 ) Here, o.
70 )= e,l.
m of 70 and 6 is 210 Oo.
6 )= 210 6 = 35 Ne o.
)= 35 To determine o.
7 ) Here, o.
70 )= e,l.
m of 70 and 7 is 70 Oo.
7 )= 70 = 10 Ne o.
7 )= To determine o.
8 ) Here, o.
70 )= e,l.
m of 70 and 8 is 280 Oo.
8 )= 280 8 = 35 Ne o.
8 )= 35 To determine o.
9 ) Here, o.
70 )= e,l.
m of 70 and 9 is 630 Oo.
9 )= 630 9 = 70 Ne o.
)= 70 Page 337 of 343 Mannan et.
Science and Technology Indonesia, 7 .
333-343 To determine o.
10 ) Here, o.
70 )= e,l.
m of 70 and 10 is 70 Oo.
10 )= 70 10 = 7 Ne o.
10 )= 7 To determine o.
24 ) Here, o.
70 )= e,l.
m of 70 and 24 is 840 Oo.
24 )= 840 24 = 35 Ne o.
24 )= 35 To determine o.
11 ) Here, o.
70 )= e,l.
m of 70 and 11 is 770 Oo.
11 )= 770 11 = 70 Ne o.
)= 70 To determine o.
25 ) Here, o.
70 )= e,l.
m of 70 and 25 is 350 Oo.
25 )= 350 25 = 14 Ne o.
)= 14 To determine o.
12 ) Here, o.
70 )= e,l.
m of 70 and 12 is 420 Oo.
12 )= 420 12 = 35 Ne o.
)= 35 To determine o.
26 ) Here, o.
70 )= e,l.
m of 70 and 26 is 910 Oo.
26 )= 910 26 = 35 Ne o.
)= 35 To determine o.
13 ) Here, o.
70 )= e,l.
m of 70 and 13 is 910 Oo.
13 )= 910 13 = 70 Ne o.
13 )= 70 To determine o.
27 ) Here, o.
70 )= e,l.
m of 70 and 27 is 1890 Oo.
27 )= 1890
= 70
Ne o.
27 )= 70 To determine o.
14 ) Here, o.
70 )= e,l.
m of 70 and 14 is 70 Oo.
14 )= 70 14 = 5 Ne o.
14 )= 5 To determine o.
28 ) Here, o.
70 )= e,l.
m of 70 and 28 is 140 Oo.
28 )= 140 28 = 5 Ne o.
28 )= 5 To determine o.
15 ) Here, o.
70 )= e,l.
m of 70 and 15 is 210 Oo.
15 )= 210 15 = 14 Ne o.
)= 14 To determine o.
29 ) Here, o.
70 )= e,l.
m of 70 and 29 is 2030 Oo.
29 )= 2030
29 = 70
Ne o.
29 )= 70 To determine o.
16 ) Here, o.
70 )= e,l.
m of 70 and 16 is 560 Oo.
16 )= 560 16 = 35 Ne o.
)= 35 To determine o.
30 ) Here, o.
70 )= e,l.
m of 70 and 30 is 210 Oo.
30 )= 210 30 = 7 Ne o.
)= 7 To determine o.
17 ) = 70 Here, o.
70 )= e,l.
m of 70 and 17 is 1190 Oo.
17 )= 1190 Ne o.
17 )= 70 To determine o.
31 ) Here, o.
70 )= e,l.
m of 70 and 31 is 2170 Oo.
31 )= 2170
31 = 70
Ne o.
)= 70 To determine o.
18 ) Here, o.
70 )= e,l.
m of 70 and 18 is 630 Oo.
18 )= 630 18 = 35 Ne o.
18 )= 35 To determine o.
32 ) Here, o.
70 )= e,l.
m of 70 and 32 is 1120 Oo.
32 )= 1120
32 = 35
Ne o.
32 )= 35 To determine o.
19 ) Here, o.
70 )= e,l.
m of 70 and 19 is 1330 Oo.
19 )= 1330
19 = 70
Ne o.
19 )= 70 To determine o.
33 ) Here, o.
70 )= e,l.
m of 70 and 33 is 2310 Oo.
33 )= 2310
33 = 70
Ne o.
33 )= 70 To determine o.
20 ) Here, o.
70 )= e,l.
m of 70 and 20 is 140 Oo.
20 )= 140 20 = 7 Ne o.
)= 7 To determine o.
34 ) Here, o.
70 )= e,l.
m of 70 and 34 is 1190 Oo.
34 )= 1190
34 = 35
Ne o.
34 )= 35 To determine o.
21 ) Here, o.
70 )= e,l.
m of 70 and 21 is 210 Oo.
21 )= 210 21 = 10 Ne o.
)= 10 To determine o.
35 ) Here, o.
70 )= e,l.
m of 70 and 35 is 490 Oo.
35 )= 490 35 = 14 Ne o.
)= 14 To determine o.
22 ) Here, o.
70 )= e,l.
m of 70 and 22 is 770 Oo.
22 )= 770 22 = 35 Ne o.
22 )= 35 To determine o.
36 ) Here, o.
70 )= e,l.
m of 70 and 36 is 1260 Oo.
35 )= 1260
35 = 35
Ne o.
36 )= 35 To determine o.
23 ) Here, o.
70 )= e,l.
m of 70 and 23 is 1610 Oo.
23 )= 1610
23 = 70
Ne o.
23 )= 70 To determine o.
37 ) Here, o.
70 )= e,l.
m of 70 and 37 is 2590 Oo.
37 )= 2590
= 70
Ne o.
37 )= 70 A 2022 The Authors.
Page 338 of 343 Mannan et.
Science and Technology Indonesia, 7 .
333-343 To determine o.
38 ) Here, o.
70 )= e,l.
m of 70 and 38 is 1330 Oo.
38 )= 1330
38 = 35
Ne o.
38 )= 35 To determine o.
52 ) Here, o.
70 )= e,l.
m of 70 and 52 is 1820 Oo.
52 )= 1820
52 = 35
Ne o.
52 )= 35 To determine o.
39 ) Here, o.
70 )= e,l.
m of 70 and 39 is 2730 Oo.
39 )= 2730
30 = 70
Ne o.
)= 70 To determine o.
53 ) Here, o.
70 )= e,l.
m of 70 and 53 is 3710 Oo.
53 )= 3710
53 = 70
Ne o.
)= 70 To determine o.
40 ) Here, o.
70 )= e,l.
m of 70 and 40 is 280 Oo.
40 )= 280 40 = 7 Ne o.
)= 7 To determine o.
54 ) Here, o.
70 )= e,l.
m of 70 and 54 is 1890 Oo.
54 )= 1890
54 = 35
Ne o.
)= 35 To determine o.
41 ) Here, o.
70 )= e,l.
m of 70 and 41 is 2870 Oo.
41 )= 2870
41 = 70
Ne o.
41 )= 70 To determine o.
55 ) Here, o.
70 )= e,l.
m of 70 and 55 is 770 Oo.
55 )= 770 55 = 14 Ne o.
55 )= 14 To determine o.
42 ) Here, o.
70 )= e,l.
m of 70 and 42 is 1470 Oo.
42 )= 1470
4 = 35
Ne o.
42 )= 35 To determine o.
56 ) Here, o.
70 )= e,l.
m of 70 and 56 is 280 Oo.
56 )= 280 56 = 5 Ne o.
56 )= 5 To determine o.
43 ) Here, o.
70 )= e,l.
m of 70 and 43 is 3010 Oo.
43 )= 3010
43 = 70
Ne o.
43 )= 70 To determine o.
57 ) Here, o.
70 )= e,l.
m of 70 and 57 is 3990 Oo.
57 )= 3990
= 70
Ne o.
57 )= 70 To determine o.
44 ) Here, o.
70 )= e,l.
m of 70 and 44 is 1540 Oo.
44 )= 1540
44 = 35
Ne o.
)= 35 To determine o.
58 ) Here, o.
70 )= e,l.
m of 70 and 58 is 2030 Oo.
58 )= 2030
58 = 35
Ne o.
)= 35 To determine o.
45 ) Here, o.
70 )= e,l.
m of 70 and 45 is 630 Oo.
630 )= 630 45 = 45 Ne o.
44 )= 14 To determine o.
59 ) Here, o.
70 )= e,l.
m of 70 and 59 is 4130 Oo.
59 )= 4130
59 = 70
Ne o.
)= 70 To determine o.
46 ) Here, o.
70 )= e,l.
m of 70 and 46 is 1610 Oo.
46 )= 1610
46 = 35
Ne o.
46 )= 35 To determine o.
60 ) Here, o.
70 )= e,l.
m of 70 and 60 is 420 Oo.
60 )= 420 60 = 7 Ne o.
60 )= 7 To determine o.
47 ) Here, o.
70 )= e,l.
m of 70 and 47 is 3290 Oo.
47 )= 3290
= 70
Ne o.
47 )= 70 To determine o.
61 ) Here, o.
70 )= e,l.
m of 70 and 61 is 4270 Oo.
61 )= 4270
61 = 70
Ne o.
61 )= 70 To determine o.
48 ) Here, o.
70 )= e,l.
m of 70 and 48 is 1680 Oo.
48 )= 1680
48 = 35
Ne o.
)= 35 To determine o.
62 ) Here, o.
70 )= e,l.
m of 70 and 62 is 2170 Oo.
62 )= 2170
62 = 35
Ne o.
62 )= 35 To determine o.
49 ) Here, o.
70 )= e,l.
m of 70 and 49 is 490 Oo.
49 )= 490 49 = 10 Ne o.
)= 10 To determine o.
63 ) Here, o.
70 )= e,l.
m of 70 and 63 is 4410 Oo.
63 )= 4410
63 = 70
Ne o.
)= 70 To determine o.
50 ) Here, o.
70 )= e,l.
m of 70 and 50 is 350 Oo.
50 )= 350 50 = 7 Ne o.
50 )= 7 To determine o.
64 ) Here, o.
70 )= e,l.
m of 70 and 64 is 2240 Oo.
64 )= 2240
64 = 35
Ne o.
64 )= 35 To determine o.
51 ) Here, o.
70 )= e,l.
m of 70 and 51 is 3570 Oo.
51 )= 3570
51 = 70
Ne o.
51 )= 70 To determine o.
65 ) Here, o.
70 )= e,l.
m of 70 and 65 is 910 Oo.
65 )= 910 65 = 14 Ne o.
65 )= 14 A 2022 The Authors.
Page 339 of 343 Mannan et.
Science and Technology Indonesia, 7 .
333-343 To determine o.
66 ) Here, o.
70 )= e,l.
m of 70 and 66 is 2310 Oo.
66 )= 2310
66 = 35
Ne o.
66 )= 35 To determine o.
8 ) Here, o.
71 )= e,l.
m of 71 and 8 is 568 Oo.
8 )= 568 8 = 71 Ne o.
8 )= 71 To determine o.
67 ) Here, o.
70 )= e,l.
m of 70 and 67 is 4690 Oo.
70 )= 4690
= 70
Ne o.
)= 70 To determine o.
9 ) Here, o.
71 )= e,l.
m of 71 and 9 is 639 Oo.
9 )= 639 9 = 71 Ne o.
)= 71 To determine o.
68 ) Here, o.
70 )= e,l.
m of 70 and 68 is 2380 Oo.
68 )= 2380
68 = 35
Ne o.
)= 35 To determine o.
10 ) Here, o.
71 )= e,l.
m of 71 and 10 is 710 Oo.
10 )= 710 10 = 71 Ne o.
)= 71 To determine o.
69 ) Here, o.
70 )= e,l.
m of 70 and 69 is 4830 Oo.
69 )= 4830
69 = 70
Ne o.
69 )= 70 To determine o.
11 ) Here, o.
71 )= e,l.
m of 71 and 11 is 781 Oo.
11 )= 781 11 = 71 Ne o.
11 )= 71 To determine o.
70 ) Here, o.
70 )= e,l.
m of 70 and 70 is 70 Oo.
70 )= 70 =1Ne o.
70 )= 1 To determine o.
12 ) Here, o.
71 )= e,l.
m of 71 and 12 is 852 Oo.
12 )= 852 12 = 71 Ne o.
12 )= 71 3 The Higher Prime Order of a Group for Multiplication Composition .
AoAngeli et al.
, 2020.
Gillibert, 2018.
Cavaleri et al.
, 2.
Find the order of every element in the multiplication group G= .
, a2 , a3 , a4 .
A A A , a71 = .
To determine o.
13 ) Here, o.
71 )= e,l.
m of 71 and 13 is 923 Oo.
13 )= 923 13 = 71 Ne o.
13 )= 71 Solution:
The identity element of the given group is a71 = e Ne o.
= 71 O o.
= 71 We know that o.
m )= myuI , where yuI = l.
m of m and n To determine o.
2 ) Here, o.
71 )= e,l.
m of 71 and 2 is 142 Oo.
2 )= 142 2 = 71 Ne o.
2 )= 71 To determine o.
3 ) Here, o.
71 )= e,l.
m of 71 and 3 is 213 Oo.
3 )= 213 3 = 71 Ne o.
)= 71 To determine o.
4 ) Here, o.
71 )= e,l.
m of 71 and 4 is 284 Oo.
4 )= 284 4 = 71 Ne o.
)= 71 To determine o.
5 ) Here, o.
71 )= e,l.
m of 71 and 5 is 355 Oo.
5 )= 355 5 = 71 Ne o.
5 )= 71 To determine o.
6 ) Here, o.
71 )= e,l.
m of 71 and 6 is 426 Oo.
6 )= 426 6 = 71 Ne o.
6 )= 71 To determine o.
7 ) Here, o.
71 )= e,l.
m of 71 and 7 is 497 Oo.
7 )= 497 = 71 Ne o.
7 )= 71 A 2022 The Authors.
To determine o.
14 ) Here, o.
71 )= e,l.
m of 71 and 14 is 994 Oo.
14 )= 994 14 = 71 Ne o.
)= 71 To determine o.
15 ) Here, o.
71 )= e,l.
m of 71 and 15 is 1065 Oo.
15 )= 1065
15 = 71
Ne o.
15 )= 71 To determine o.
16 ) Here, o.
71 )= e,l.
m of 71 and 16 is 1136 Oo.
16 )= 1136
16 = 71
Ne o.
16 )= 71 To determine o.
17 ) Here, o.
71 )= e,l.
m of 71 and 17 is 1207 Oo.
17 )= 1207
= 71
Ne o.
17 )= 71 To determine o.
18 ) Here, o.
71 )= e,l.
m of 71 and 18 is 1278 Oo.
18 )= 1278
18 = 71
Ne o.
)= 71 To determine o.
19 ) Here, o.
71 )= e,l.
m of 71 and 19 is 1349 Oo.
19 )= 1349
19 = 71
Ne o.
)= 71 To determine o.
20 ) Here, o.
71 )= e,l.
m of 71 and 20 is 1420 Oo.
20 )= 1420
20 = 71
Ne o.
20 )= 71 To determine o.
21 ) Here, o.
71 )= e,l.
m of 71 and 21 is 1491 Oo.
21 )= 1491
21 = 71
Ne o.
21 )= 71 Page 340 of 343 Mannan et.
Science and Technology Indonesia, 7 .
333-343 To determine o.
22 ) Here, o.
71 )= e,l.
m of 71 and 22 is 1562 Oo.
22 )= 1562
22 = 71
Ne o.
22 )= 71 To determine o.
36 ) Here, o.
71 )= e,l.
m of 71 and 36 is 2556 Oo.
36 )= 2556
36 = 71
Ne o.
36 )= 71 To determine o.
23 ) Here, o.
71 )= e,l.
m of 71 and 23 is 1633 Oo.
23 )= 1633
23 = 71
Ne o.
)= 71 To determine o.
37 ) Here, o.
71 )= e,l.
m of 71 and 37 is 2627 Oo.
37 )= 2627
= 71
Ne o.
)= 71 To determine o.
24 ) Here, o.
71 )= e,l.
m of 71 and 24 is 1704 Oo.
24 )= 1704
24 = 71
Ne o.
)= 71 To determine o.
38 ) Here, o.
71 )= e,l.
m of 71 and 38 is 2698 Oo.
38 )= 2698
38 = 71
Ne o.
)= 71 To determine o.
25 ) Here, o.
71 )= e,l.
m of 71 and 25 is 1775 Oo.
25 )= 1775
25 = 71
Ne o.
25 )= 71 To determine o.
39 ) Here, o.
71 )= e,l.
m of 71 and 39 is 2769 Oo.
39 )= 2769
39 = 71
Ne o.
39 )= 71 To determine o.
26 ) Here, o.
71 )= e,l.
m of 71 and 26 is 1846 Oo.
26 )= 1846
26 = 71
Ne o.
26 )= 71 To determine o.
40 ) Here, o.
71 )= e,l.
m of 71 and 40 is 2840 Oo.
40 )= 2840
40 = 71
Ne o.
40 )= 71 To determine o.
27 ) Here, o.
71 )= e,l.
m of 71 and 27 is 1917 Oo.
27 )= 1917
= 71
Ne o.
27 )= 71 To determine o.
41 ) Here, o.
71 )= e,l.
m of 71 and 41 is 2911 Oo.
41 )= 2911
41 = 71
Ne o.
41 )= 71 To determine o.
28 ) Here, o.
71 )= e,l.
m of 71 and 28 is 1988 Oo.
28 )= 1988
28 = 71
Ne o.
)= 71 To determine o.
42 ) Here, o.
71 )= e,l.
m of 71 and 42 is 2982 Oo.
42 )= 2982
42 = 71
Ne o.
)= 71 To determine o.
29 ) Here, o.
71 )= e,l.
m of 71 and 29 is 2059 Oo.
29 )= 2059
29 = 71
Ne o.
)= 71 To determine o.
43 ) Here, o.
71 )= e,l.
m of 71 and 43 is 3053 Oo.
43 )= 3053
43 = 71
Ne o.
)= 71 To determine o.
30 ) Here, o.
71 )= e,l.
m of 71 and 30 is 2130 Oo.
30 )= 2130
30 = 71
Ne o.
30 )= 71 To determine o.
44 ) Here, o.
71 )= e,l.
m of 71 and 44 is 3124 Oo.
44 )= 3124
44 = 71
Ne o.
44 )= 71 To determine o.
31 ) Here, o.
71 )= e,l.
m of 71 and 31 is 2201 Oo.
31 )= 2201
31 = 71
Ne o.
31 )= 71 To determine o.
45 ) Here, o.
71 )= e,l.
m of 71 and 45 is 3195 Oo.
45 )= 3195
45 = 71
Ne o.
45 )= 71 To determine o.
32 ) Here, o.
71 )= e,l.
m of 71 and 32 is 2272 Oo.
32 )= 2272
32 = 71
Ne o.
32 )= 71 To determine o.
46 ) Here, o.
71 )= e,l.
m of 71 and 46 is 3266 Oo.
46 )= 3266
46 = 71
Ne o.
)= 71 To determine o.
33 ) Here, o.
71 )= e,l.
m of 71 and 33 is 2343 Oo.
33 )= 2343
33 = 71
Ne o.
)= 71 To determine o.
47 ) Here, o.
71 )= e,l.
m of 71 and 47 is 3337 Oo.
47 )= 3337
= 71
Ne o.
)= 71 To determine o.
34 ) Here, o.
71 )= e,l.
m of 71 and 34 is 2414 Oo.
34 )= 2414
34 = 71
Ne o.
34 )= 71 To determine o.
48 ) Here, o.
71 )= e,l.
m of 71 and 48 is 3408 Oo.
48 )= 3408
48 = 71
Ne o.
48 )= 71 To determine o.
35 ) Here, o.
71 )= e,l.
m of 71 and 35 is 2485 Oo.
35 )= 2485
35 = 71
Ne o.
35 )= 71 To determine o.
49 ) Here, o.
71 )= e,l.
m of 71 and 49 is 3479 Oo.
49 )= 3479
49 = 71
Ne o.
49 )= 71 A 2022 The Authors.
Page 341 of 343 Science and Technology Indonesia, 7 .
333-343 Mannan et.
To determine o.
50 ) Here, o.
71 )= e,l.
m of 71 and 50 is 3550 Oo.
50 )= 3550
50 = 71
Ne o.
50 )= 71 To determine o.
64 ) Here, o.
71 )= e,l.
m of 71 and 64 is 4544 Oo.
64 )= 4544
64 = 71
Ne o.
64 )= 71 To determine o.
51 ) Here, o.
71 )= e,l.
m of 71 and 51 is 3621 Oo.
51 )= 3621
51 = 71
Ne o.
)= 71 To determine o.
65 ) Here, o.
71 )= e,l.
m of 71 and 65 is 4615 Oo.
65 )= 4615
65 = 71
Ne o.
)= 71 To determine o.
52 ) Here, o.
71 )= e,l.
m of 71 and 52 is 3692 Oo.
52 )= 3692
52 = 71
Ne o.
)= 71 To determine o.
66 ) Here, o.
71 )= e,l.
m of 71 and 66 is 4686 Oo.
66 )= 4686
66 = 71
Ne o.
66 )= 71 To determine o.
53 ) Here, o.
71 )= e,l.
m of 71 and 53 is 3763 Oo.
53 )= 3763
53 = 71
Ne o.
53 )= 71 To determine o.
67 ) Here, o.
71 )= e,l.
m of 71 and 67 is 4757 Oo.
67 )= 4757
= 71
Ne o.
)= 71 To determine o.
54 ) Here, o.
71 )= e,l.
m of 71 and 54 is 3834 Oo.
54 )= 3834
54 = 71
Ne o.
54 )= 71 To determine o.
68 ) Here, o.
71 )= e,l.
m of 71 and 68 is 4828 Oo.
68 )= 4828
68 = 71
Ne o.
68 )= 71 To determine o.
55 ) Here, o.
71 )= e,l.
m of 71 and 55 is 3905 Oo.
55 )= 3905
55 = 71
Ne o.
55 )= 71 To determine o.
69 ) Here, o.
71 )= e,l.
m of 71 and 69 is 4830 Oo.
69 )= 4899
69 = 71
Ne o.
69 )= 71 To determine o.
56 ) Here, o.
71 )= e,l.
m of 71 and 56 is 3976 Oo.
56 )= 3976
56 = 71
Ne o.
)= 71 To determine o.
70 ) Here, o.
71 )= e,l.
m of 71 and 70 is 4970 Oo.
70 )= 4970
= 71
Ne o.
)= 71 To determine o.
57 ) Here, o.
71 )= e,l.
m of 71 and 57 is 4047 Oo.
57 )= 4047
= 71
Ne o.
)= 71 To determine o.
71 ) Here, o.
71 )= e,l.
m of 71 and 71 is 71 Oo.
71 )= 71 =1Ne o.
71 )= 1 To determine o.
58 ) Here, o.
71 )= e,l.
m of 71 and 58 is 4118 Oo.
58 )= 4118
58 = 71
Ne o.
58 )= 71
CONCLUSIONS
To determine o.
59 ) Here, o.
71 )= e,l.
m of 71 and 59 is 4189 Oo.
59 )= 4189
59 = 71
Ne o.
59 )= 71 To determine o.
60 ) Here, o.
71 )= e,l.
m of 71 and 60 is 4260 Oo.
60 )= 4260
60 = 71
Ne o.
60 )= 71 To determine o.
61 ) Here, o.
71 )= e,l.
m of 71 and 61 is 4331 Oo.
61 )= 4331
61 = 71
Ne o.
)= 71 To determine o.
62 ) Here, o.
71 )= e,l.
m of 71 and 62 is 4402 Oo.
62 )= 4402
62 = 71
Ne o.
62 )= 71 To determine o.
63 ) Here, o.
71 )= e,l.
m of 71 and 63 is 4473 Oo.
63 )= 4473
63 = 71
Ne o.
63 )= 71 A 2022 The Authors.
We hope that this work will be useful for group theory related to the higher order of an element of a group.
Our result is the order of every element of a group in different types of the higher-order group.
This result has found extensive use in statistics, information theory and geometrics, etc.
Then all expected results in this paper will help us to understand a better solution to complicated order.
ACKNOWLEDGMENT
I would like to thank my respectable teacher Prof.
Dr.
Nurul Alam Khan, for guidance throughout the research process.
REFERENCES