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Home / Modern Abstract Algebra / Prove that a^n = e iff m | n( m divides n).

Prove that a^n = e iff m | n( m divides n).

Theorem: Let G be a group with identity e. If a ∊ G and O(a) = m, then for some positive integer n, an = e if and only if m | n (m divides n).

Proof: O(a) = m

⇒ m is the least positive integer such that am = e. Hence an = e.

⇒ n ≥ m.

By division algorithm we can write n = h m +r,

where h is an integer and r is a non negative integer less than m.

Now e = an

                 = ahm+r

                     = ahm ar

                   = (am)h ar

            = eh ar

         = e ar

                                                                        = ar

We have ar = e with 0 ≤ r < m.

Since m is the smallest positive integer such that am = e.

We must have r = 0.

Thus n = hm which implies that m is a factor of n.

i.e., m | n.

Conversely, let m | n. Then n hm for some integer h,

and so an = ahm

                    = (am)h

               = eh

               = e.

Thus the theorem is proved.

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