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Asked Sep 17, 2019
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64

64. Let a and b belong to a group. If lal and lbl are relatively prime,
show that (a) n (b) = {e}.
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64. Let a and b belong to a group. If lal and lbl are relatively prime, show that (a) n (b) = {e}.

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Expert Answer

Step 1

Under the given conditions, to show that the cyclic groups generated by a and b have only common element, namely the identity

Step 2

Note that H is a subgroup , as it is the intersection of two subgroups <a> and <b>

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Let H=< a>n<b> Now,<a> the cyclic subgroup has order a <b>=the cyclic subgroup has order b

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Step 3

conclude first that <a> intersection <b> ...

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According to Lagrange's theorem |H (number of elements order of H) divides both aand b| But, aland b are relatively prime (no common divisors except 1) |H 1

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