Prove the following.(1) Z ∗ 5 is a cyclic group. (2) Z ∗ 8 is not a cyclic group.

he298c 2021-02-27 Answered

Prove the following.
(1) Z×5 is a cyclic group.
(2) Z×8 is not a cyclic group.

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Expert Answer

Isma Jimenez
Answered 2021-02-28 Author has 84 answers

1) We know that Zn={a:where gcd(a,n)=1,1an} form a froup under multiplication with inverse bar1. Now,
Z5={a:where gcd(a,5)=1,1a5}={1,2,3,4}.
Check that
(2)4=(4)2=1,
(42=16≡∈mod5)
As here 4 is the element positive integer such that 24=1, therefore, order of 2Z5=4. Hence, Z5 is cyclic
2) Again Z8={a:where gcd(a,8)=1,1a8}={1,3,5,7}.,
but 3¯2=5¯2=7¯2=1¯
(32=91mod8, similar approach for 5,7). So, there does not exists any element of order 4Z8, hence Z8 is not cyclic.
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