The order of x is same as the order of x inverse.sagews

In the excercises you have been asked to prove that if $G$ is a group and $x \in G$ then $|x|=|x^{-1}|$. Recall $|x|$ is the least integer $0 < n$ and $x^{n} = e$ (The textbook is using $x^{n} = 1$).

Below you will find some calculations in $\mathbb{Z}_n$ and permutaions group that suggest it is a true result.

H = SymmetricGroup(5)
#for r in H:
    #s = r^-1
    #print r, " order is: ", r.multiplicative_order() , " r^-1=", s," order: ", r^-1.multiplicative_order()

#A pedantic, but efficient way
elements = H.list()
for r in H:
    s = r^-1
    print r, "\t order is: ", r.order() , " r^-1=", s,"\t order: ", s.order()
    del elements[elements.index(s)] #Remove r^-1, just to reduce computations

() order is: 1 r^-1= () order: 1 (1,2) order is: 2 r^-1= (1,2) order: 2 (1,2,3,4,5) order is: 5 r^-1= (1,5,4,3,2) order: 5 (1,3,4,5) order is: 4 r^-1= (1,5,4,3) order: 4 (2,3,4,5) order is: 4 r^-1= (2,5,4,3) order: 4 (1,3,5,2,4) order is: 5 r^-1= (1,4,2,5,3) order: 5 (1,2,4)(3,5) order is: 6 r^-1= (1,4,2)(3,5) order: 6 (1,3,4,5,2) order is: 5 r^-1= (1,2,5,4,3) order: 5 (1,3,5)(2,4) order is: 6 r^-1= (1,5,3)(2,4) order: 6 (1,4,2,5,3) order is: 5 r^-1= (1,3,5,2,4) order: 5 (1,4)(2,3,5) order is: 6 r^-1= (1,4)(2,5,3) order: 6 (1,3)(2,5,4) order is: 6 r^-1= (1,3)(2,4,5) order: 6 (2,4)(3,5) order is: 2 r^-1= (2,4)(3,5) order: 2 (1,4,3)(2,5) order is: 6 r^-1= (1,3,4)(2,5) order: 6 (1,5,4,3,2) order is: 5 r^-1= (1,2,3,4,5) order: 5 (1,4)(2,5,3) order is: 6 r^-1= (1,4)(2,3,5) order: 6 (1,5,3)(2,4) order is: 6 r^-1= (1,3,5)(2,4) order: 6 (1,4)(3,5) order is: 2 r^-1= (1,4)(3,5) order: 2 (1,4,2,3,5) order is: 5 r^-1= (1,5,3,2,4) order: 5 (1,5,3,2,4) order is: 5 r^-1= (1,4,2,3,5) order: 5 (1,3,2,5,4) order is: 5 r^-1= (1,4,5,2,3) order: 5 (1,5,4,2) order is: 4 r^-1= (1,2,4,5) order: 4 (1,2,5,4,3) order is: 5 r^-1= (1,3,4,5,2) order: 5 (1,4,3,2,5) order is: 5 r^-1= (1,5,2,3,4) order: 5 (2,5,4,3) order is: 4 r^-1= (2,3,4,5) order: 4 (1,5,4,2,3) order is: 5 r^-1= (1,3,2,4,5) order: 5 (1,4,3,2) order is: 4 r^-1= (1,2,3,4) order: 4 (1,4,2)(3,5) order is: 6 r^-1= (1,2,4)(3,5) order: 6 (1,5,4,3) order is: 4 r^-1= (1,3,4,5) order: 4 (1,5,3,2) order is: 4 r^-1= (1,2,3,5) order: 4 (1,5,2,4,3) order is: 5 r^-1= (1,3,4,2,5) order: 5 (2,5,3) order is: 3 r^-1= (2,3,5) order: 3 (4,5) order is: 2 r^-1= (4,5) order: 2 (2,3) order is: 2 r^-1= (2,3) order: 2 (1,5) order is: 2 r^-1= (1,5) order: 2 (2,4,3) order is: 3 r^-1= (2,3,4) order: 3 (1,3,2) order is: 3 r^-1= (1,2,3) order: 3 (3,4) order is: 2 r^-1= (3,4) order: 2 (2,5,4) order is: 3 r^-1= (2,4,5) order: 3 (1,5,4)(2,3) order is: 6 r^-1= (1,4,5)(2,3) order: 6 (1,5,2) order is: 3 r^-1= (1,2,5) order: 3 (1,5,3) order is: 3 r^-1= (1,3,5) order: 3 (1,5)(2,4,3) order is: 6 r^-1= (1,5)(2,3,4) order: 6 (1,4,2) order is: 3 r^-1= (1,2,4) order: 3 (1,4,3) order is: 3 r^-1= (1,3,4) order: 3 (1,5,4) order is: 3 r^-1= (1,4,5) order: 3 (1,2,4,3) order is: 4 r^-1= (1,3,4,2) order: 4 (1,2)(4,5) order is: 2 r^-1= (1,2)(4,5) order: 2 (1,4) order is: 2 r^-1= (1,4) order: 2 (1,2,3,5) order is: 4 r^-1= (1,5,3,2) order: 4 (1,2,5,3) order is: 4 r^-1= (1,3,5,2) order: 4 (1,2)(3,4) order is: 2 r^-1= (1,2)(3,4) order: 2 (1,5)(2,3) order is: 2 r^-1= (1,5)(2,3) order: 2 (3,4,5) order is: 3 r^-1= (3,5,4) order: 3 (1,2,4,5) order is: 4 r^-1= (1,5,4,2) order: 4 (1,3) order is: 2 r^-1= (1,3) order: 2 (2,3)(4,5) order is: 2 r^-1= (2,3)(4,5) order: 2 (1,2,5) order is: 3 r^-1= (1,5,2) order: 3 (1,5)(3,4) order is: 2 r^-1= (1,5)(3,4) order: 2 (2,4) order is: 2 r^-1= (2,4) order: 2 (2,3,4) order is: 3 r^-1= (2,4,3) order: 3 (1,2,3) order is: 3 r^-1= (1,3,2) order: 3 (1,2,5,4) order is: 4 r^-1= (1,4,5,2) order: 4 (2,5) order is: 2 r^-1= (2,5) order: 2 (1,4,5) order is: 3 r^-1= (1,5,4) order: 3 (1,2,3,4) order is: 4 r^-1= (1,4,3,2) order: 4 (1,2,3)(4,5) order is: 6 r^-1= (1,3,2)(4,5) order: 6 (1,5)(2,3,4) order is: 6 r^-1= (1,5)(2,4,3) order: 6 (1,4,5)(2,3) order is: 6 r^-1= (1,5,4)(2,3) order: 6 (1,5,2,3,4) order is: 5 r^-1= (1,4,3,2,5) order: 5 (1,2,3,5,4) order is: 5 r^-1= (1,4,5,3,2) order: 5 (1,3,4) order is: 3 r^-1= (1,4,3) order: 3 (1,4,5,2) order is: 4 r^-1= (1,2,5,4) order: 4 (2,4,5) order is: 3 r^-1= (2,5,4) order: 3 (1,2,5)(3,4) order is: 6 r^-1= (1,5,2)(3,4) order: 6 (1,2)(3,4,5) order is: 6 r^-1= (1,2)(3,5,4) order: 6 (1,3,2,4,5) order is: 5 r^-1= (1,5,4,2,3) order: 5 (1,3,2,5) order is: 4 r^-1= (1,5,2,3) order: 4 (1,5,2)(3,4) order is: 6 r^-1= (1,2,5)(3,4) order: 6 (1,2,4,3,5) order is: 5 r^-1= (1,5,3,4,2) order: 5 (1,3,5) order is: 3 r^-1= (1,5,3) order: 3 (1,3,4,2) order is: 4 r^-1= (1,2,4,3) order: 4 (1,3,4)(2,5) order is: 6 r^-1= (1,4,3)(2,5) order: 6 (1,3)(2,4,5) order is: 6 r^-1= (1,3)(2,5,4) order: 6 (1,5,2,3) order is: 4 r^-1= (1,3,2,5) order: 4 (1,2,4) order is: 3 r^-1= (1,4,2) order: 3 (1,3,2)(4,5) order is: 6 r^-1= (1,2,3)(4,5) order: 6 (2,3,5) order is: 3 r^-1= (2,5,3) order: 3 (1,3)(4,5) order is: 2 r^-1= (1,3)(4,5) order: 2 (1,4)(2,5) order is: 2 r^-1= (1,4)(2,5) order: 2 (1,3,5,2) order is: 4 r^-1= (1,2,5,3) order: 4 (2,5)(3,4) order is: 2 r^-1= (2,5)(3,4) order: 2 (3,5) order is: 2 r^-1= (3,5) order: 2 (2,4,5,3) order is: 4 r^-1= (2,3,5,4) order: 4 (1,3)(2,5) order is: 2 r^-1= (1,3)(2,5) order: 2 (1,2,5,3,4) order is: 5 r^-1= (1,4,3,5,2) order: 5 (1,5,3,4) order is: 4 r^-1= (1,4,3,5) order: 4 (1,3,5,4) order is: 4 r^-1= (1,4,5,3) order: 4 (2,3,5,4) order is: 4 r^-1= (2,4,5,3) order: 4 (1,4,3,5) order is: 4 r^-1= (1,5,3,4) order: 4 (1,3)(2,4) order is: 2 r^-1= (1,3)(2,4) order: 2 (1,2,4,5,3) order is: 5 r^-1= (1,3,5,4,2) order: 5 (2,4,3,5) order is: 4 r^-1= (2,5,3,4) order: 4 (1,4,5,2,3) order is: 5 r^-1= (1,3,2,5,4) order: 5 (1,5,2,4) order is: 4 r^-1= (1,4,2,5) order: 4 (1,3,4,2,5) order is: 5 r^-1= (1,5,2,4,3) order: 5 (1,3,2,4) order is: 4 r^-1= (1,4,2,3) order: 4 (1,2)(3,5) order is: 2 r^-1= (1,2)(3,5) order: 2 (1,4,5,3,2) order is: 5 r^-1= (1,2,3,5,4) order: 5 (2,5,3,4) order is: 4 r^-1= (2,4,3,5) order: 4 (1,4,2,5) order is: 4 r^-1= (1,5,2,4) order: 4 (1,4,3,5,2) order is: 5 r^-1= (1,2,5,3,4) order: 5 (1,4,5,3) order is: 4 r^-1= (1,3,5,4) order: 4 (1,4)(2,3) order is: 2 r^-1= (1,4)(2,3) order: 2 (1,3,5,4,2) order is: 5 r^-1= (1,2,4,5,3) order: 5 (1,5)(2,4) order is: 2 r^-1= (1,5)(2,4) order: 2 (1,5,3,4,2) order is: 5 r^-1= (1,2,4,3,5) order: 5 (1,4,2,3) order is: 4 r^-1= (1,3,2,4) order: 4 (3,5,4) order is: 3 r^-1= (3,4,5) order: 3 (1,2)(3,5,4) order is: 6 r^-1= (1,2)(3,4,5) order: 6

G = IntegerModRing(2*3*7*5) #We will study multiplicative group. The result is trivail in additive group
units = [G(i) for i in range(210) if gcd(i, 210) == 1] #We reserve a space for elements that has an inverse

for r in  units:
    print r, "\t order is: ", r.multiplicative_order() , " r^-1=", r^-1," order: ", G(r^-1).multiplicative_order()
    del units[units.index(G(r^-1))] #Remove r^-1, just to reduce computations

1 order is: 1 r^-1= 1 order: 1 13 order is: 4 r^-1= 97 order: 4 17 order is: 12 r^-1= 173 order: 12 19 order is: 6 r^-1= 199 order: 6 23 order is: 12 r^-1= 137 order: 12 29 order is: 2 r^-1= 29 order: 2 37 order is: 12 r^-1= 193 order: 12 41 order is: 2 r^-1= 41 order: 2 47 order is: 12 r^-1= 143 order: 12 53 order is: 12 r^-1= 107 order: 12 59 order is: 6 r^-1= 89 order: 6 61 order is: 6 r^-1= 31 order: 6 71 order is: 2 r^-1= 71 order: 2 79 order is: 6 r^-1= 109 order: 6 83 order is: 4 r^-1= 167 order: 4 101 order is: 6 r^-1= 131 order: 6 103 order is: 12 r^-1= 157 order: 12 113 order is: 4 r^-1= 197 order: 4 121 order is: 3 r^-1= 151 order: 3 127 order is: 4 r^-1= 43 order: 4 149 order is: 6 r^-1= 179 order: 6 163 order is: 12 r^-1= 67 order: 12 181 order is: 2 r^-1= 181 order: 2 191 order is: 6 r^-1= 11 order: 6

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