if a in G (group ) such as o(a)=mn prove the existence of g and h...

if a in G (group ) such as o(a)=mn

prove the existence of g and h in G such as a=gh=hg and o(g)=m o(h)=n

Solutions

Expert Solution

Answer:-

Given that

First, let a=g and b=h be the elements whose orders are mm and nn, respectively. I guessed that we can find the element of order lcm(m,n)l explicitly, instead of simply proving its existence. Furthermore, I also guessed that the element can be expressed in the form akblakbl, because the statement must also hold when G is generated by a and b.

order (a)= mn = ab =gh

then ,

we attempet for the values in those solution,

then

Hence,prove that the existence of g and h in G such as a=gh=hg and o(g)=m o(h)=n

where a=h ,b=g and order (a)=ab =gh=mn

Colby Messinger answered 1 year ago

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