Answer to Question #139354 in Abstract Algebra for J

Question #139354
Assume that (G,*) is a group, that g is an element of G and that t is the smallest positive integer such that g^t=e

prove that g^n=e if and only if t |n. use division algorithm and explain why r must = 0
1
Expert's answer
2020-10-22T17:55:08-0400

Let us prove the converse first.

If tnt|n . thus there exist kZk\in \Bbb Z such that kt=nkt=n .

Now,

gn=gkt=(gt)k=ek=eg^n=g^{kt}=(g^t)^k=e^k=e

Thus,

gn=eg^n=e

Suppose, tt not divides nn , hence by division algorithm there are p,r such that


n=pt+r,0<r<tn=pt+r,\hspace{1cm}0<r<t

Moreover we have given

gn=e    gn=gpt+r    (gt)pgr    epgr    egr=gr=eg^n=e\\ \implies g^n=g^{pt+r}\\ \implies(g^t)^pg^r\\ \implies e^pg^r\\ \implies eg^r=g^r=e

But we have given that tt is given smallest , hence by definition tr    rtt|r\implies r\ge t , hence we get a contradiction.

Therefore r must 0, hence

n=pt    tnn=pt\implies t|n

We are done.


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